Thermacoustic piezoelectric generator

ABSTRACT

An electroactive transducer converts between acoustical power and electrical power. The transducer includes a diaphragm and a perimeter member. The perimeter member includes at least one electroactive element and is mechanically coupled to the perimeter of the diaphragm such that displacement of the diaphragm stresses the electroactive element.

REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 10/810,907, filed Mar. 26, 2004, which claims priority to U.S.Provisional Patent Application 60/459,541, filed Mar. 31, 2003, theentire content of both of which is incorporated herein by reference.

GOVERNMENT SPONSORSHIP

This work was supported by the Office of Naval Research grant numberN00014-98-1-0212 and N00014-03-0652, and the Department of EnergyFreedomCAR and Vehicle Technologies Program, through cooperativeagreement number DE-FC26-04NT42113. Accordingly the United Statesgovernment may have certain rights in this invention.

FIELD OF THE INVENTION

The invention relates to the field of electrical power generation fromheat using oscillatory linear motion alternators.

BACKGROUND OF THE INVENTION

Oscillatory heat engines such as thermoacoustic engines or Stirlingengines may be used for refrigerators or heat pumps, where the energy ofgas pressure and volume oscillations is converted into the movement ofheat into the engine at low temperatures and out of the engine at hightemperatures. Oscillatory heat engines may also be used as prime movers,where heat flow into the engine at high temperatures and out of theengine at low temperature is converted into gas pressure and volumeoscillations.

Conventional Stirling engines have long been used as prime movers thataccept heat at a high temperature, reject less heat at lowertemperature, and convert a large fraction of the difference into usefulwork in the form of pressure and volume oscillations of a working gas,such as helium. They are able to do this with high efficiency, definedas the useful work output divided by the hot heat input. Heat isexchanged between the external world and the working gas through a hotand a cold heat exchanger. A regenerator is placed between these twoheat exchangers. The regenerator is a fine porous structure, such as astack of fine metal screen, that is in intimate thermal contact with thegas. Conventional Stirling engines use two sliding pistons connected bya mechanical linkage to effect translations of the working gas throughthe regenerator that are properly phased with volume changes of the gas.In a common version of such engines a displacer piston translates thegas through the regenerator and a power piston takes net power from theengine by allowing expansion of the gas while the pressure is high andcontraction of the gas while the pressure is low. Although Stirlingengines have been under development for almost 200 years, theirmechanical complexity has historically limited their commercialviability. Stirling engines with high power density typically haveinternal pressures in excess of 20 atmospheres, creating problems withthe pressure vessel penetrations and sliding seals necessary forconnecting rods and moving pistons. The mass of the moving parts alsolimits the operating frequency of the engine to the detriment of theengine's power density. The development of the free-piston Stirlingengine eliminated the mechanical linkage between the power and displacerpistons and some of the sliding seals by resonating the pistons on gasor mechanical springs. However, the problems associated with internalsliding piston seals and the limits brought on by large piston massesremain.

The pressure and volume changes of the working gas can be consideredmore generally to be manifestations of sound in the gas. This acousticpoint of view inspired the relatively recent invention of thethermoacoustic-Stirling engine described in U.S. Pat. Nos. 4,114,380;4,355,517; 6,032,464 and 6,314,740 and WIPO publications WO 99/20957 andWO 02/086445 all hereby incorporated by reference for their teachings ofsuch engines. In the acoustic point of view, the function of theflywheel in the original Stirling engine is accomplished instead byresonance—the repetitive oscillation of energy back and forth betweenpotential and kinetic forms. Potential energy is stored in thecompression or expansion of stiffness elements which may take severalforms, such as metal springs or gaseous volumes that act as springs.Kinetic energy is stored in the oscillatory motion of masses, also ofseveral forms, such as the motion of a mass of a fluid or a mass ofsolids. Resonance may be established between discrete lumped stiffnessand mass elements, as in the case of a solid block on a coil metalspring or of the so-called Helmholtz acoustic resonator; or resonancesmay be established among distributed elements, each element having bothstiffness and mass-like properties, as in the case of a ringing tuningfork or the acoustic resonances of a gas filled tube; or resonance mayoccur in a continuum between these lumped and distributed extremes.

The thermoacoustic-Stirling engine shares with the free-piston Stirlingengine the use of gas springs, but its much simplified mechanical designemploys acoustic networks to form acoustic masses and springs that mimicthe function and motion of mechanical pistons, thereby completelyeliminating all pistons, mechanical linkages and sliding seals. In someforms of the thermoacoustic-Stirling engine, a resonant tube of gas,approximately the length of half a wavelength of sound, functions as thepower piston.

An earlier version of thermoacoustic engine utilizes standing waves andimperfect thermal conduction between the working gas and the porousmedium, called a stack in this type of engine rather than a regenerator,to achieve the proper phasing between gas motion through the stack andexpansions and contractions of the gas. Examples of patents teachingthis type of technology are U.S. Pat. Nos. 4,398,398; 4,489,553;4,722,201 hereby incorporated by reference. Thermoacoustic standing waveengines share with thermoacoustic-Stirling engines the advantages ofeliminating the moving pistons of conventional or free-piston Stirlingdevices, and are even simpler in their construction. However, theysuffer from poor efficiency associated with the necessarily poor heattransfer between the gas and the stack. Like manythermoacoustic-Stirling engines, they also use a long resonating tube ofgas to establish a resonance.

Many other oscillatory heat engines utilizing resonance exist as well,some examples of which are a cascaded thermoacoustic device, (U.S. Pat.No. 6,658,862 hereby incorporated by reference), a no-stackthermoacoustic device, (R. S. Wakeland and R. M. Keolian,“Thermoacoustics with Idealized Heat Exchangers and No Stack,” J.Acoust. Soc. Am., 111 (6), Pt. 1, 2654-2664, 2002, hereby incorporatedby reference), a heat-controlled acoustic wave system of Marrison (U.S.Pat. No. 2,836,033 hereby incorporated by reference) and some very oldheat engines such as singing flames, the Sondhauss tube and the Rijketube, described by Rayleigh (J. W. S. Rayleigh, “The Theory of Sound,”Dover, N.Y., 1945, Vol. 2, pp. 224-234, hereby incorporated byreference).

The output power per unit volume (the volumetric power density) of athermoacoustic engine may be increased by operating the engine at higherfrequencies. The power output of a thermoacoustic-Stirling or standingwave engine is proportional to the engine area normal to the directionof sound, the mean pressure of the working gas, the speed of sound ofthe working gas, and the square of the ratio of the acoustic pressureamplitude to mean pressure of the working gas. The output power does notexplicitly depend on the operating frequency of the engine. However, theoverall length of the engine is proportional to the wavelength of sound,which is inversely proportional to the operating frequency, even whensolids are used in place of gasses to establish resonance. Because thevolume of the engine is proportional to the normal area times the enginelength, it is therefore possible and advantageous to reduce the volumetaken up by the engine by raising the operating frequency, which can bedone with little penalty in output power. Raising the operatingfrequency is limited by the ability of the heat exchangers to functionproperly when their effective size is limited by the shorter acousticdisplacements that result at higher frequencies, by the parasiticthermal conduction between hot and cold regions of the engine which willbe closer together as the frequency is increased, and by the moving massof any transducers used to exchange electrical power with the engine.

Although it has long been the practice of designers of conventional andfree piston Stirling engines to use the location of the power piston forthe transduction of power into or out of the heat engine, this was notoriginally the practice in thermoacoustics. An early attempt to shortena thermoacoustic device was made by Hofler and Grant (L. A. Grant,“Investigation of the Physical Characteristics of a Mass ElementResonator,” M.S. Thesis, Naval Postgraduate School, Monterey, Calif.,1992, National Technical Information Service ADA251792) by substitutinga resonating mass for the mass impedance presented by the typical nearlyhalf wavelength long resonating tube. (Impedance, or more properly thespecific acoustic impedance, is the complex ratio of pressure tovelocity. A mass impedance has the pressure leading the velocity by 90degrees so that the pressure is in phase with the acceleration, as wouldbe the case for a mass.) Similarly, U.S. Pat. No. 6,314,740 and WIPOpublication WO 99/20957, for the case of thermoacoustic-Stirlingengines, and U.S. Pat. No. 6,578,364 (herein incorporated by reference),for the cases of both the thermoacoustic standing wave andthermoacoustic-Stirling engines, teach that when a thermoacoustic engineis used with a transducer for removing or adding power (onlyelectrodynamic examples are shown), the length of the engine can begreatly shortened by substituting the moving mass of the transducer forthe acoustic mass impedance of the working gas in the half wavelengthlong resonant tube. In effect, the transducer is used as the powerpiston of the conventional or free piston Stirling engines.

Thus, transducers that present a mass impedance may be beneficial; sincethe core of oscillatory heat engines often present an impedance which isprimarily stiffness-like (pressure lagging velocity by nearly 90degrees), the combination of mass-like transducers and stiffness-likeengines, along with other mass and stiffness impedances as desired, maybe combined to form a resonant system that result in compact useful heatengines.

The only examples of transducers that have been given in the prior art,however, that present a mass impedance which may be used for thisbeneficial shortening of the engine are electrodynamic, because in theprior art only electrodynamic transducers have a stroke (peak to peakdisplacement amplitude) sufficient to accomplish this task. Linearmotion electrodynamic transducers (e.g. U.S. Pat. Nos. 4,623,808 and5,389,844 hereby incorporated by reference) may be used for driving orgenerating electricity with any of the oscillatory heat engine types.This electrodynamic class of transducers, when used as alternators, forexample, use various topologies to induce an electromotive force in awire by way of a changing magnetic flux through a stationary coil, or byway of wire motion through a static magnetic field. Electrodynamicalternators, however, have the disadvantage that their moving mass tendsto be large. This is not generally a problem at low operatingfrequencies, where it is more important that the alternator have a largestroke to roughly match the large displacement amplitude of the sound,but the large moving mass limits the use of electrodynamic transducersat high frequencies.

A number of piezoelectric generators have been proposed in the patentliterature for a variety of purposes unrelated to the generation ofelectricity from oscillating heat engines: a vibrating reed driven bythe passage of air in the nose cone of a missile (U.S. Pat. No.4,005,319 hereby incorporated by reference), bender elements lining aautomobile muffler to pull energy out of the sound of auto exhaust,(U.S. Pat. No. 4,467,236 hereby incorporated by reference),piezoelectric elements embedded in motor vehicle tires to pull energyout of the flexing of tires via slip rings (U.S. Pat. No. 4,504,761hereby incorporated by reference), a stack of piezoelectric elementsexcited by the pressure pulse of an internal combustion engine (U.S.Pat. No. 4,511,818 hereby incorporated by reference), a strip ofpiezoelectric plastic implanted into the human body to energizeimplanted electronics with body movement (U.S. Pat. No. 5,431,694 herebyincorporated by reference), bender elements that are plucked by camswhich move in response to ocean waves (U.S. Pat. No. 5,814,921 herebyincorporated by reference), rotary motion of an eccentric shaft thatapplies oscillating stresses onto piezoelectric elements in contact withthe eccentric (U.S. Pat. No. 6,194,815 hereby incorporated byreference), or which are inertially stressed by the eccentric motion(U.S. Pat. No. 6,429,576 hereby incorporated by reference). Some ofthese applications appear to be impractical.

The use of piezoelectric alternators in place of electrodynamicalternators in a thermoacoustic application has been suggested for usewith the standing wave engines (W. C. Ward et al., “Thermoacousticengine scaling, acoustic and safety study,” Los Alamos NationalLaboratory unclassified report LA-12103-MS, 1991). Their alternatorconfiguration is interesting and clever. However, like many traditionalacoustic transducers, it presents to the resonator a very highimpedance, and it has a very limited available stroke, factors which areassociated with the high mechanical stiffness and limited mechanicalstrain of a raw piece of piezoelectric ceramic. The high impedance andlow stroke of their transducer forces it to be placed near the pressureanti-node of the sound field where the acoustic velocity is small. Sinceduring a cycle of sound energy oscillates between potential and kineticforms and the transducer is incapable of holding much of the kineticenergy due to its limited stroke, the resonator needs to be about aslong as a half wavelength of sound so that a large volume of fast movinggas near the velocity anti-node may accept the kinetic energy. This LosAlamos configuration of a piezoelectric alternator cannot take the spacesaving advantage of using the alternator as a resonating mass that maysubstitute for much of the gaseous mass of the acoustic resonator.

An earlier patent (U.S. Pat. No. 3,822,388 hereby incorporated byreference) showing a simple stack of piezoelectric ceramic coupled withhydraulic fluid and a column of mercury to the pressure oscillationsgenerated by an otherwise conventional Stirling engine, also presentsthe engine with a transducer that is very stiff. With its high impedanceand large mass of coupling fluids, it is unsuitable for use withthermoacoustic engines.

In addition to being used as prime movers, thermoacoustic and Stirlingheat engines may be used in the opposite sense, accepting work in theform of the coupled pressure and volume changes of sound in order topull heat from a thermal load at low temperature and reject heat at ahigher temperature, useful for constructing refrigerators or heat pumps.Transducers used as acoustic drivers to convert electrical power intothe acoustic power needed to run the refrigerators or heat pumps wouldpotentially have the same space saving advantages as their alternatorcounterparts if they were made to present a mass impedance.Additionally, by combining a prime mover with a refrigerator, it ispossible to make heat driven refrigerators, for example for use inremote or mobile applications where connection to the electrical powergrid is not feasible or desirable. It is often necessary, however, tomake electricity available for the running of fans and pumps for thedistribution of the cooling effect or for auxiliary uses. It istherefore desirable to have an efficient means of generating electricityin heat driven refrigerators, chillers and heat pumps. It is alsodesirable to have compact transducers (alternators or drivers) that canoperate at high frequency that present a mass rather than a stiffnessimpedance, which may be used to establish resonance with oscillatoryheat engines presenting a primarily stiffness impedance. These needs areaddressed by the present invention.

SUMMARY OF THE INVENTION

The present invention provides a thermoacoustic-piezoelectric electricalpower generator that uses the heat driven pressure and volumeoscillations of thermoacoustic prime movers to power piezoelectricalternators. Refrigeration stages may be added to the generator as well.A first embodiment of the present invention is a thermoacousticgenerator, including a housing containing a working volume of gas with apressure. A thermoacoustic core is supported in the housing and includesa first heat exchanger and a second heat exchanger. The thermoacousticcore is operable to introduce acoustical power into the housing, therebyoscillating the pressure of the gas at a frequency. A piezoelectricalternator is supported in the housing and has a face that is movablewhen acted on by acoustical power. The alternator further includes aportion of piezoelectric material operable to produce electrical powerwhen acted on by a stress. The portion of piezoelectric material is inmechanical communication with the movable face, such that the movementof the face stresses the portion of piezoelectric material so as toproduce electrical power. The alternator has a moving mass that servesas a substantial portion of the resonating mass inside the housing. Thisprovides a pressure oscillation frequency in the housing that issubstantially lower than for a similar system with a rigid memberreplacing the alternator. In a preferred version of the firstembodiment, the movable face of the alternator substantially blocks thepassage of gas.

In some versions of the first embodiment, the movable face is a firstdiaphragm. The housing has a sidewall, and the diaphragm may have aperimeter seal substantially sealing the diaphragm to the sidewall ofthe housing. The perimeter seal may be selected from the groupconsisting of a roll sock, a bellows, and a clearance seal. Thegenerator may also include a second diaphragm forming a second face ofthe alternator. The portion of piezoelectric material is also inmechanical communication with the second diaphragm, and is disposedbetween the first and second diaphragms.

In some versions of the first embodiment, the piezoelectric alternatorportion includes a perimeter member that includes the portion ofpiezoelectric material. The perimeter member is configured such thatcompression of the perimeter member causes compression of the portion ofpiezoelectric material. The perimeter member surrounds a central area. Ahub is disposed in the central area and is movable relative to theperimeter member along the axis. The hub is in mechanical communicationwith the movable face of the alternator. A plurality of spokesinterconnect the hub and the perimeter member such that relativemovement of the hub along the axis compresses the perimeter member andthereby compresses the piezoelectric material. This version of the firstembodiment may have a housing with a sidewall and a first diaphragmserving as the alternator face. A perimeter seal seals the firstdiaphragm to the sidewall of the housing, and may be selected from thegroup consisting of a roll sock, a bellows, and a clearance seal. It mayfurther include a second diaphragm in mechanical communication with thehub, with the second diaphragm also including perimeter seals sealingthe diaphragm to the sidewall of the housing. The perimeter member, thehub and spokes are disposed between the first and second diaphragms.

In another version of the first embodiment, the perimeter member isgenerally ring-shaped. In this version, the piezoelectric materialportion of the ring-shaped perimeter member may comprise substantiallyall of the ring-shaped perimeter member.

In yet another version, the perimeter member is generallypolygonal-shaped with intersection zones defined between adjacentgenerally straight segments. The portion of piezoelectric materialcomprises a portion of each of the straight segments. The spokes may beinterconnected with intersection zones of the polygonal-shaped member.Also, the generally straight segments of the perimeter member may eachfurther include a spring in series with the portion of piezoelectricmaterial.

A different version of a piezoelectric alternator for the firstembodiment includes a perimeter support member generally defining analternator plane, with the support member surrounding a central area. Ahub is disposed in the central area, with the hub being movable relativeto the perimeter support member along an axis generally perpendicular tothe plane. The hub is in mechanical communication with a movablealternator face. The portion of piezoelectric material comprises aplurality of piezoelectric bimorph members each having an inner end inmechanical communication with the hub and an outer end supported by theperimeter support member, such that relative movement of the hub alongthe axis flexes the bimorph members. The bimorph members may begenerally wedge-shaped such that their width parallel to the alternatorplane is narrower at the inner ends than at the outer ends. Theperimeter support member may be generally circular. The housing has aside wall and the alternator face may be a first diaphragm with aperimeter seal sealing the first diaphragm of the side wall of thehousing, the perimeter seal being selected from the group consisting ofa roll sock, a bellows, and a clearance seal. The generator may furtherinclude a second diaphragm in mechanical communication with the hub. Thesecond diaphragm may include a perimeter seal sealing the diaphragm atthe sidewall of the housing, with the perimeter support member, the hub,and the bimorph members being disposed between the first and seconddiaphragms.

In yet another version of the first embodiment, the piezoelectricalternator may further include at least one spring in series with aportion of piezoelectric material, so as to alter the stiffness of thepiezoelectric alternator.

Another version of a piezoelectric alternator for the first embodimentincludes a perimeter wall having a plurality of wall segmentsinterconnected by springs. The portion of piezoelectric materialcomprises at least a portion of one of the wall segments. The wallsegments have a surface that serves as the movable face of thealternator. The alternator may further comprise an alternator bodyenclosing a portion of the working volume of gas with the perimeter wallforming part of the alternator body. Alternatively, the perimeter wallmay substantially separate the housing into first and second coaxialregions, with the thermoacoustic cores supported in one of the regions.A second thermoacoustic core may be supported in the other region withthe thermoacoustic cores being coaxially arranged. According to afurther alternative, the piezoelectric material may comprisesubstantially the entirety of all of the wall segments of thepiezoelectric alternator.

A second embodiment of the present invention is directed to apiezoelectric transducer for converting between acoustical power,consisting of pressure and velocity, and electrical power, consisting ofpotential and current. The transducer includes the perimeter member withat least one portion of piezoelectric material. The perimeter member isconfigured such that compression of the perimeter member causescompression of the portion of piezoelectric material. The perimetermember surrounds a central area. The hub is disposed in the centralarea, with the hub being movable relative to the perimeter member alongan axis. A plurality of spokes interconnect the hub and the perimetermember such that relative movement of the hub along the axis compressesthe perimeter member and thereby compresses the piezoelectric material.

In some versions of the second embodiment, the transducer serves as adriver that converts electrical power to acoustical power. Thetransducer may further include a first diaphragm in mechanicalcommunication with the hub such that the movement of hub moves at leasta portion of the diaphragm. Electric power is applied to thepiezoelectric material causing movement of at least a portion of theperimeter member, which causes movement of the spokes, which causesmovement of the hub, which causes movement of at least a portion of thediaphragm, thereby creating acoustical power. Some versions furtherinclude a second diaphragm in mechanical communication with the hub,such that movement of the hub moves at least a portion of the seconddiaphragm, and the hub and spokes are disposed between the first andsecond diaphragm.

In another version of the second embodiment, the transducer is analternator that converts acoustical power to electrical power. Thetransducer further includes a first diaphragm in mechanicalcommunication with the hub, such that movement of the hub moves at leasta portion of the diaphragm. When acoustical power is applied to thefirst diaphragm, it causes movement of the face and hub, therebycompressing the piezoelectric material. The transducer may furtherinclude a second diaphragm in mechanical communication with the hub,such that movement of the hub moves at least a portion of the seconddiaphragm, and the hubs and spokes are disposed between the first andsecond diaphragms.

In some versions of the second embodiment, the perimeter member isgenerally ring-shaped. The piezoelectric material portion of thering-shaped perimeter member may comprise substantially the entirering-shaped perimeter member. In other versions of the secondembodiment, the perimeter member is generally polygonal-shaped withintersection zones defined between adjacent generally straight segments.The piezoelectric material portion of the perimeter member comprises aportion of each of the straight segments. The spokes may beinterconnected with intersection zones of the polygonal-shaped perimetermember. Also, the straight segments of the perimeter member may eachfurther include a spring in series with a portion piezoelectricmaterial.

According to a third embodiment of the present invention, apiezoelectric transducer is provided for converting between acousticalpower, consisting of pressure and velocity, and electrical power,consisting of potential and current. The transducer includes a perimetersupport member generally defining a transducer plane. The membersurrounds a central area. The hub is disposed in the central area, withthe hub being movable relative to the perimeter support member along anaxis generally perpendicular to the plane. A plurality of piezoelectricbimorph members each have an inner end in mechanical communication withthe hub and an outer end supported by the perimeter support member, suchthat relative movement of the hub along the axis flexes the bimorphmembers. In the third embodiment, the transducer may be an alternatoroperable to convert acoustical power to electrical power. It may includea first diaphragm in mechanical communication with the hub, such thatmovement of the face causes movement of the hub along the axis. Thebimorph members may be generally wedge-shaped such that the width of themembers parallel to the transducer plane is narrower at the inner endsthan at the outer ends.

According to a forth embodiment of the present invention, apiezoelectric transducer is provided for converting between acousticalpower, consisting of pressure and velocity, and electrical power,consisting of potential and current. The transducer includes atransducer assembly with at least one piezoelectric element configuredto produce electrical power when acted upon by a mechanical force. Thetransducer also includes at least one spring in series with thepiezoelectric element so as to alter the resonant frequency of thetransducer assembly.

According to a fifth embodiment of the present invention, athermoacoustic device includes a housing containing a working volume ofgas with a pressure. A piezoelectric transducer separates the housinginto a first area containing a first volume of gas and a second areacontaining a second volume of gas. The transducer comprises a perimeterwall having at least one portion of piezoelectric material and at leastone spring in series. A first thermoacoustic core is supported in thefirst area of the housing and includes a pair of heat exchangers. Asecond thermoacoustic core is supported in the second area of thehousing and includes a pair of heat exchangers.

According to a further embodiment of the present invention, anelectroactive transducer is provided for converting between acousticalpower, consisting of pressure and velocity, and electrical power,consisting of potential and current. The transducer includes a diaphragmhaving a perimeter and a central portion. The diaphragm has a neutralposition and at least one flexed position wherein the central portion isdisplaced axially relative to the neutral position. The perimeter andthe central portion are disposed generally in a common diaphragm planewhen the diaphragm is in the neutral position. A perimeter memberincludes at least one electroactive element. The perimeter membergenerally defines a transducer plane and surrounds a central area. Theperimeter member is mechanically coupled to the perimeter of thediaphragm such that displacement of the central portion from the neutralposition to the flexed position stresses the electroactive element. Insome embodiments, the diaphragm plane is generally coextensive with thetransducer plane and the diaphragm is disposed in a central area definedby the perimeter member such that the perimeter member generallysurrounds the diaphragm. The electroactive element may comprise aplurality of electroactive elements disposed generally in a ring and theelectroactive elements may be piezoelectric elements.

In yet another embodiment of the present invention, an electroactivetransducer is provided for converting between acoustical power,consisting of pressure and velocity, and electrical power, consisting ofpotential and current. The transducer has a first face and an opposedsecond face. A diaphragm has a first convex face defining at least partof the first face of the transducer and an opposed concave face definingat least part of the second face of the transducer. The diaphragm has aperimeter and a central portion. The diaphragm has a neutral positionand at least one flexed position wherein the central portion isdisplaced axially relative to the neutral position. A perimeter memberincludes at least one electroactive element and generally defines atransducer plane surrounding a central area. The perimeter member ismechanically coupled to the perimeter of the diaphragm such thatdisplacement of the central portion from the neutral position to theflexed position stresses the electroactive element. In some versions,the perimeter of the diaphragm is disposed in the transducer plane whenthe diaphragm is disposed in the central area defined by the perimetermember such that the perimeter member generally surrounds the diaphragm.The at least one electroactive element may comprise a plurality ofelectroactive elements disposed generally in a ring, and theelectroactive elements may be piezoelectric elements.

In a further embodiment of the present invention, a thermoacousticgenerator, refrigerator or heat pump is provided that includes a housingcontaining a working volume of gas with a pressure. A thermoacousticengine is supported in the housing and has a first heat exchanger and asecond heat exchanger. The thermoacoustic engine is operable tointroduce acoustical power into the housing or to remove acousticalpower from the housing. An electroactive transducer is supported in thehousing and includes a diaphragm having a perimeter and a centralportion that is movable when acted on by acoustical power. The diaphragmhas a neutral position and at least one flexed position wherein thecentral portion is displaced axially relative to the neutral position. Aperimeter member includes at least one electroactive element andgenerally defines a transducer plane and surrounds a central area. Theperimeter member is mechanically coupled to the perimeter of thediaphragm such that displacement of the central portion from the neutralposition to the flexed position stresses the electroactive element.

In yet a further embodiment of the present invention, an acoustic deviceis provided that includes a housing having a volume defined therein. Thehousing is filled with a gas having a mean density, and an adiabaticsound speed. An electroactive transducer is at least partially disposedin the housing. The transducer includes a diaphragm having a perimeterand a central portion. The diaphragm has a neutral position and at leastone flexed position wherein the central portion is displaced axiallyrelative to the neutral position. A perimeter member includes at leastone electroactive element and generally defines a transducer plane andsurrounds a central area. The perimeter member is mechanically coupledto the perimeter of the diaphragm such that displacement of the centralportion from the neutral position to the flexed position stresses theelectroactive element. The diaphragm divides the housing such that afirst volume is defined on one side of the diaphragm and a second volumeis defined on the other side of the diaphragm. The diaphragm has aradius a The diaphragm has a tensile force per length of edge T givenby, wherein F_(R) is the arithmetic sum of all radial time dependenttensile forces that the perimeter member applies to the diaphragm and T₀is a bias tension present when the diaphragm is in the neutral position.The diaphragm is configured such that χ is in the range of 7 to 34inclusive, wherein χ is defined as χ=πρ₀c_(a) ²a⁴/(V_(o)T), wherein V₀is the effective volume of the gas spring compressed by the diaphragm.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a partially cutaway side view of a balanced configurationfor a thermoacoustic-Stirling piezoelectric generator according to afirst embodiment of the present invention. The generator is sized todeliver 4 kW of electricity derived from the waste heat of diesel truckexhaust.

FIG. 2A shows a cross-sectional view of a quadratically drivenpiezoelectric ring transducer/alternator that forms one aspect of thepresent invention, taken along lines 2A-2A of FIG. 2B. Axial motion ofthe hub flexes the spokes which compresses the piezoelectric ring.

FIG. 2B shows a cross-sectional view of the ring transducer/alternatorof FIG. 2A, taken along lines 2B-2B of FIG. 2A.

FIG. 3 shows operation of the piezoelectric ring transducer/alternatorthrough one cycle of sound. There are two compressions of thepiezoelectric ring, at 90° and 270°, for each acoustic cycle.

FIG. 4A shows a partially cutaway perspective view of a 2.0 kWthermoacoustic Stirling piezoelectric generator core.

FIG. 4B shows a partially cutaway side view of the generator of FIG. 4A.

FIG. 5 shows a cross-sectional view of an alternative embodiment of aquadratically driven piezoelectric ring transducer/alternator includingazimuthal springs for lowering the ring resonant frequency.

FIG. 6A shows a slice of the ring alternator of FIG. 2A for applicationof Newton's second law.

FIG. 6B shows a slice of the ring alternator of FIG. 5 for applicationof Newton's second law.

FIG. 7A shows an equivalent circuit for the ring alternator of FIG. 5.

FIG. 7B shows a reduced equivalent circuit for the ring alternator ofFIG. 5 when reactive elements may be ignored.

FIG. 8 shows a graph plotting ring alternator efficiency vs.R_(L)|R_(L MAX EFFICIENCY).

FIG. 9 shows a cross-sectional view of an alternator design with rollsocks as a means for providing flexible seals.

FIG. 10 shows a cross-sectional view of an alternator design withbellows as a means for providing flexible seals.

FIG. 11 shows a cross-sectional view of an alternator design withclearance seals as a means for providing a flexible seals.

FIG. 12 shows an alternative embodiment of a piezoelectric ringtransducer/alternator, wherein the piezoelectric ceramic is in the formof a striped cylinder.

FIG. 13 shows an alternative version of the ring transducer of FIG. 12,having wire spokes with coils, attached to a keystone-like support, witha polygonal piezoelectric ceramic ring made up of straight stacks ofceramic elements.

FIG. 14 shows a detail of an alternate means for attaching spokes to astriped ring piezoelectric ceramic.

FIG. 15 shows a detail of another alternate means for attaching spokesto a striped ring piezoelectric ceramic.

FIG. 16 shows a cross-sectional view of one of the azimuthal springsused in the alternator of FIG. 5.

FIG. 17 shows a detail of a straight spoke end in plan view.

FIG. 18 shows a detail of a tapered spoke end in plan view.

FIG. 19 is an exploded view of an alternative embodiment of apiezoelectric ring alternator, wherein the alternator is based on aresonant ring without spokes.

FIG. 20 shows a cross-sectional view of another alternative embodimentof a piezoelectric generator according to the present invention.

FIG. 21 shows a cross-sectional view of yet another alternativeembodiment of a piezoelectric generator according to the presentinvention.

FIG. 22 shows a cross-sectional view of an alternative version of thegenerator of FIG. 21.

FIG. 23 shows a partially cutaway side view of athermoacoustic/piezoelectric generator that also functions as a chiller;

FIG. 24 is a perspective view of an embodiment of a flexible diaphragmtransducer according to another aspect of the present invention.

FIG. 25 is a perspective cutaway view of a portion of the transducer ofFIG. 24.

FIG. 26 is a cutaway perspective view of another version of a flexiblediaphragm transducer.

FIGS. 27A-27D are detailed views of an end detail for radial slots.

FIGS. 28A-28E are simplified drawings of an embodiment of a flexiblediaphragm transducer shown in five different positions.

FIG. 29 is a graph of voltage versus displacement for an exemplarytransducer such as shown in FIGS. 28A-28E.

FIG. 30 is a cutaway perspective view of yet a further embodiment of aflexible diaphragm transducer.

FIGS. 31A-31C are simplified drawings of another version of a flexiblediaphragm transducer, having a domed diaphragm, and showing thediaphragm in three positions.

FIG. 32 is a partially cutaway perspective view of an alternativeversion of a flexible diaphragm transducer.

FIGS. 33A-33C are detailed views of portions of arches which may be usedto seal radial and axial slots in one of the transducers.

DETAILED DESCRIPTION OF THE INVENTION

The present invention is directed to a heat driventhermoacoustic-piezoelectric generator that uses high efficiency,compact, reliable thermoacoustic-Stirling engines coupled to a linearlyoscillating alternator based on piezoelectric rather than electrodynamictechnology. The alternator moving mass replaces the acoustic orresonating mass of the traditional thermoacoustic resonator. An optionalchiller or heat pump stage may be added to the generator. Thepiezoelectric alternators presented here may be used with otheroscillatory engines as well. The piezoelectric alternators may also beused in the reverse sense as driving transducers for the creation ofacoustic energy from electricity.

Electric power generation using thermoacoustic-Stirling engines coupledto piezoelectric alternators should allow significant advantages insize, weight, radiated noise, maintenance, and efficiency when comparedto other technologies. In addition, this technology is well suited toaccept heat from a variety of sources, such as waste heat, a dedicatedburner, the sun, or nuclear sources. A first preferred embodiment forsuch a generator is shown at 10 in the configuration of FIG. 1. Twopiezoelectric alternators 12 and 14, which act as resonating masses,separate a pressure vessel 16 into upper 18, central 20 and lower 22chambers, which act as energy generating gas springs. The pressurevessel 16 is filled with a working gas, for example helium gas at 10atmospheres of mean pressure (other pressures could easily be used).Each alternator 12 and 14 has two faces that move together along theaxis of the pressure vessel. Pressure swings in the chambers are coupledto the volume swept out by the alternator face motion. The faces aresealed to the pressure vessel 16 by a means for allowing motion, such asconstructing the faces in the form of a corrugated diaphragm, although alight weight piston sealed to the vessel with a bellows or roll-sock, ora clearance seal may also be used (see FIGS. 9-11). A small controlledleak (not shown) allows the working gas of the engines to equilibratewith the space inside the alternators over a long time scale.

Four thermoacoustic-Stirling engines 24, 26, 28 and 30 in the chamberscreate pressure swings in the gas, for example with 1 atmosphereamplitude. The oscillations are driven thermally by external heatapplied to the engines through hot heat exchangers and rejected heattaken away by cold heat exchangers, as described in the next paragraph.The engines are a very compact coaxial adaptation of the highlyefficient thermoacoustic-Stirling engines recently pioneered by thethermoacoustics group at Los Alamos National Laboratory (see U.S. Pat.No. 6,032,464 hereby incorporated by reference). The pressure swingsdrive the two piezoelectric alternators 180° out of phase—thealternators move in unison toward or away from each other—in avibrationally balanced configuration. The overall topology is similar toa full wavelength acoustic wave, except that the effective wavelength isconsiderably shortened by substituting the moving mass of thealternators for the acoustic or resonating mass of the sound wave at thevelocity anti-nodes (see U.S. Pat. Nos. 6,578,364, 6,314,740 and WO99/20,957 and WO 02/086,445 incorporated herein by reference).

The thermoacoustic engines 24-30 are preferably substantially identical.Therefore, only the first engine 24 will be described in more detail. Itshould be noted that FIG. 1 is a cutaway side view of the generator 10and that the pressure vessel 16, the engines 24-30, and the generators12 and 14 are preferably generally radially symmetrical with each itembeing generally disc-shaped or generally cylindrical. The engine 24includes a generally tubular nacelle 32 that extends coaxially with thewall of the pressure vessel 16 and is spaced from the wall 16 so as toprovide an annular space 34 between the nacelle 32 and the wall 16. Athermoacoustic core 36 is contained inside the nacelle 32. The core 36includes a cold heat exchanger 38 and a hot heat exchanger 40sandwiching a regenerator 42.

The thermoacoustic engine generates sound in the following manner. Theannular space between the resonator wall 16 and the nacelle 32 acts as alumped acoustic mass. The space 44 between the cold heat exchanger andthe end of the resonator acts as a gas spring. This mass-springsub-system is tuned to have a resonant frequency well above theoperating frequency of, for example, 400 Hz, causing the pressureoscillations at the closed end of the resonator to be slightlyresonantly enhanced with respect to its driving point near the diaphragmof the alternator 12, with negligible phase shift. Thus the acousticpressure is nearly in phase on both sides of the regenerator 42 but theamplitude is slightly higher on the cold side than on the hot side. Whenthe diaphragm of the alternator 12 moves towards the regenerator 42 andthe pressures on the hot and cold sides swing positive, this slightpressure difference across the regenerator drives the gas in theregenerator to move from colder to hotter locations. The gas in theregenerator, being in intimate thermal contact with the regeneratorscreen, expands as it moves towards hotter locations. The gas expansionduring the positive pressure phase does work on the rest of the gas,which causes the positive pressure to swing even higher than it wouldhave otherwise been, and pushes back on the diaphragm giving it energy.During the other half of the acoustic cycle, when the diaphragm movesaway from the regenerator 42, the pressures on the hot and cold sidesswing negative (lower pressure), with a slightly larger swing on thecold side. This causes a pressure gradient which drives the gas in theregenerator to move toward the cold side, where it cools and contracts,thereby causing the pressure to swing even further negative, which pullsthe diaphragm back, again adding energy to the diaphragm.

Between the hot and cold heat exchangers 40 and 38 is the regenerator 42which may be a stack of fine metal screen. Thermoacoustic-Stirlingengines are similar to conventional mechanical and free-piston Stirlingengines in that the working gas situated within the regeneratorundergoes an approximation to the ideal Stirling thermodynamic cycle.However, the thermoacoustic version of the Stirling engine eliminatesthe sliding mechanical displacer and power pistons of conventional orfree-piston engines by using acoustic networks, which is expected toimprove engine reliability and cost. In the generator of FIG. 1, gas inthe annular space 34 between the nacelle 32 surrounding the engine core36 and the pressure vessel wall 16 acts somewhat like the displacerpiston while the alternator acts as the power piston of the conventionalor free-piston Stirling engine.

The first Los Alamos engine operated between a hot temperature T_(H) of725° C. and cold temperature T_(C) of 15° C., while producing 710 W ofsound and achieving a 30 percent thermal to acoustic efficiency. Thisefficiency is 42 percent of the ideal Carnot efficiency 1−T_(C)/T_(H),which is 71 percent at these temperatures. The higher the operatingfrequency, the smaller this type of engine can be made with littleconsequence to the output power. The first Los Alamos engine was large,however, operating at around 80 Hz for convenience of construction.Their engine was designed using their program DeltaE, which embodies thegood but still imperfect physics knowledge base for thermoacoustics.DeltaE is usually accurate to within 20 percent when care is taken toincluded all known high amplitude nonlinear effects, and is publiclyavailable.

The thermoacoustic-Stirling engines used in the present invention are amore compact coaxial geometry than used by Los Alamos. These engineshave been modeled with the Los Alamos DeltaE software. The DeltaE modelpredicts that each engine should achieve 1.4 kW of acoustic power at 400Hz, with 33 percent thermal to acoustic conversion efficiency, runningbetween 650° C. and 30° C. heat exchanger metal temperatures, for anefficiency relative to Carnot of 49 percent.

Referring again to FIG. 1, two thermoacoustic-Stirling engines aresituated on either side of an alternator. They may generate 10 percentamplitude (15 PSI), 400 Hz pressure oscillations in the helium. Theoscillations are 180° out of phase on either side of the alternator,driving the alternator faces along the generator axis. The resonance ofthe combined system can be understood in terms of the central mass ofthe alternator oscillating between the gas springs of each engine; eachalternator and pair of engines is nominally a spring-mass-spring system.

The heart of each engine consists of a hot heat exchanger, aregenerator, and a cold heat exchanger. The regenerator may simply be astack of fine wire screen. Hot combustion products from a burner (notshown) or other heat source circulate into, through, and out of the hotheat exchanger via tubular penetrations of the pressure vessel. In theembodiment of FIG. 1, hot combustion products travel through a mainexhaust pipe 46. Hot inlet tubes 48 allow hot combustion products toflow from the main exhaust pipe 46 to each of the hot heat exchangers,such as 40. Hot outlet tubes 50 allow the hot combustion products toflow from the hot heat exchangers to a secondary exhaust pipe 52. Aswill be clear to those of skill in the art, the heat may be provided tothe hot heat exchangers in a variety of ways, including provision of hotgas from any source, or hot liquid. In one embodiment of the presentinvention, hot combustion products flow through the main hot heatexchanger 40, causing the metal temperature of the hot heat exchanger toreach 1200° F. (650° C.). Inlet coolant line 54 provides cold heattransfer fluid to cold inlet tubes 56, which communicate with the coldheat exchanger, such as 36. Cold outlet tubes 58 allow the cold heattransfer fluid to flow from the cold heat exchangers to an outletcooling line 60. Again, cold gas or fluid may be provided to the coldheat exchangers in a variety of ways and from a variety of sources.Also, as will be clear to those of skill in the art, the terms hot andcold, as used herein, are relative terms. In one embodiment of thepresent invention, heat transfer fluid circulating through the cold heatexchanger via its pressure vessel penetrations keep the cold exchangernear ambient temperature, 90° F. (32° C.). The heat exchanger pressurevessel penetrations and the arrangement of the engine componentssuspended within the pressure vessel are chosen to minimize the adversethermal expansion stresses associated with the hot heat exchangertemperature.

Surrounding the regenerator 42 and heat exchangers 38 and 40 is thenacelle 32 that splits the gas into a central, axial part and an outerannular part. A filler ring may also be used to help guide the gas flow.The gas in the annular passage 34 outside the nacelle 32 acts somewhatlike the displacer piston of a conventional mechanical Stirling engine,while the diaphragm takes the role of the power piston. As shown, thenacelle 32 may be offset somewhat such that the cold heat exchanger 36is close to one end of the nacelle, while the other end of the nacelleextends past the hot heat exchanger 40 to an open end 62. The gas alongthe central axis within the nacelle 32, between its open end 62 and thehot heat exchanger 40, acts as a thermal buffer tube. This is a layer ofgas that provides thermal isolation between the hot heat exchanger, therest of the gas in the engine and the alternator face. The nacelle 32 ispreferably constructed from thin material, is hollow and is filled withthermal insulation to minimize the heat leak through the nacelle betweenthe hot heat exchanger and the gas in the annular space outside thenacelle. Additionally, the inside surface of the nacelle preferably isconstructed of a thin, low thermal conductivity material, such as rolledstainless steel shim stock, in order to minimize the heat leak along thenacelle between the hot heat exchanger and the cold heat exchanger. Theouter surface of the nacelle 32 may be constructed of a thin higherthermal conductivity material, such as aluminum, brass, copper or steel,and thermally anchored to the cold heat exchanger, so as to control thetemperature of the gas in the annular passage outside the nacelle. Anexpansion joint, sliding joint, or flexible seal may be provided nearthe open end of the nacelle to allow for thermal expansion of its innersurface relative to its outer surface in response to hot heat exchangertemperatures.

This relatively simple engine structure causes the quiescent state ofthe helium gas to become unstable, as described above, without requiringsliding parts or pistons. When the hot heat exchanger is brought to asufficiently high temperature, the gas spontaneously undergoes pressureoscillations that drive the alternator, as previously described. Thespecific design of the alternator used in the power generation system ofthe invention can vary widely. Importantly, the alternator must have thecapability of translating the pressure oscillations generated by theacoustic engine(s) into stress upon a piezoelectric element(s) thusgenerating power. Three example alternator designs will now bediscussed.

EXAMPLE 1 Ring Alternator

A first preferred embodiment of the alternator is the quadraticallydriven piezoelectric ring transducer or alternator 64 of FIGS. 2A and2B. The transducer consists of a ring 66 of piezoelectric material 68,such as the poled ceramic lead-zirconate-titanate PZT, attached to setsof flexible radial spokes 70 via keystone-like supports 71. Otherpiezoelectric materials may be used, as will be clear to those of skillin the art. The ring 66 can be circular or polygonal; a hexagon with sixsets of spokes is shown in the figure. Other shapes are also possible.The ring 66 may also be referred to as a perimeter member that includesa portion of piezoelectric material. The member is configured such thatcompression of the perimeter member causes compression of thepiezoelectric portion. In the illustrated embodiment, the piezoelectricmaterial forms substantially all of the straight segments extendingbetween the supports 71, located at the intersection zones betweenadjacent segments.

For best performance the PZT preferably is organized as a stack ofplates excited in their 3-3 mode as shown. Alternatively, the PZTsections may be plates extending from support 71 to support 71, excitedin the 3-1 mode with electrodes on their transverse faces, for lesscost. The spokes connect to a central hub 72, which moves along itsaxis. In this embodiment, the ring or perimeter member 66 may be said togenerally define an alternator or transducer plane, and the hub movesalong an axis generally perpendicular to the plane. The spokes 70 lie inor are generally parallel to the plane.

The hub connects to the moving faces of the alternator, here shown ascorrugated diaphragms 74. The diaphragms may take a variety of forms.The illustrated diaphragms 74 are generally flat, but it may bepreferable to make them cone shaped for stiffness. The faces are actedon by acoustical power, thereby moving the faces. Mounting springs 76are shaped leaf springs that are stiff in the axial direction butcompliant in the radial direction. They hold the internal componentsaxially while allowing compression in the piezoelectric material.

FIG. 3 shows the operation of the alternator through one acoustic cycle,in a sequence from 0° to 360°. As acoustic pressure swings cause thealternator faces or diaphragms 74 and hub 72 to deflect in eitherdirection along the axis (at 90° or 270°), the spokes 70 bend, comeunder tension, and pull the ring 66 inward causing the ring toexperience compression. Thus, there are two cycles of ring compressionfor every cycle of sound. This frequency doubling allows the use of halfas much PZT for the same output power.

Because it is a stiff ceramic, PZT requires mechanical power withrelatively high forces and small motions, whereas the engine naturallydelivers mechanical power with relatively low forces and large motion.Two force amplifying mechanisms are being used here to overcome thismechanical impedance mismatch. One is the conversion of axial force onthe hub 72 into radial tension in the spokes 70, the second is theconversion of tension in the spokes 70 into compression of the ring 66.

Preferably, the elements of the piezoelectric material are polarizedalong the direction of the stack, but with every other element pointingin the opposite direction. Electrodes of adjacent elements touch, andevery other electrode junction is wired together. In this way, thepiezoelectric elements are mechanically in series but are allelectrically in parallel. The electrical output of the elements is thenrectified and fed to a power inverter/controller, as needed, to regulatethe output voltage and frequency of the electrical power. Theinverter/controller may also be responsible for regulating the fuel flowto a burner, if this is the desired source of heat, to match the powerneeds of the generator as the electrical load changes. Further detailspertaining to the construction of this example of the alternator aredisclosed in Example 4, below.

EXAMPLE 2 Bimorph Alternator

A second embodiment of a piezoelectric alternator 78 that presents amass impedance to the thermoacoustic engines, thus reducing thenecessary size of the pressure-vessel/acoustic-resonator, is shown inthe two cutaway views of FIGS. 4A and 4B, drawn generally to scale. Fordefiniteness, the figures show the core of a 2.0 kWthermoacoustic-Stirling piezoelectric generator. In one version, thesteel pressure-vessel/acoustic-resonator 80 is 36 cm (14″) long, 15 cm(6″) in diameter at the ends, and bulges to 23 cm (9″) diameter at thecenter. The resonator is filled with 10 atmosphere (150 psi) of heliumgas. The walls of the resonator are 0.65 mm thick in the small diameterregions and 1.0 mm thick near the center to perform safely as a pressurevessel. Other dimensions, working fluids and pressures, of course, arepossible.

Two flexible diaphragms 82 and 84 are located in the center of theresonator. In this embodiment, the diaphragms form the movable faces ofthe alternator. The centers of the diaphragms are constrained by a post86 to move together in the direction of the resonator axis. The outerrims of the diaphragms form a seal against the resonator wall 80. Inthis version, seventy-five piezoelectric disks 88 are flexed by thediaphragms through a mechanical connection at the post. These diaphragmsand piezoelectric disks comprise the electrical transducer, oralternator, portion of the generator. Each of the 75 disks is made oftwelve triangular piezoelectric bimorph cantilever wedges 90. Only onesuch disk of wedges is shown in the figures. The triangular wedges 90have a clamped boundary condition at their outer diameter, and a simplysupported boundary condition at the post, produced by a small flexiblestrut at the vertex of each wedge. These boundary conditions cause auniform bending stress throughout the bimorph, maximizing the effectiveuse of the piezoelectric material. This arrangement also minimizes themoving mass, thereby allowing a high operating frequency, which reducesthe overall size of the resonator. The bimorphs in this embodiment are a0.65 mm thick sandwich of two thin PZT sheets glued on either face of athin metal sheet. The force amplification necessary for impedancematching the alternator 78 to the engines is accomplished by therelatively small forces applied to the bimorph vertex causing relativelylarge bending stresses in the PZT. The bimorphs are made thin enoughthat the maximum allowable stress in the PZT ceramic is not exceededwhen the engines drive the alternator faces to their maximum amplitudewhile the generator is operating at full power. Enough piezoelectricdisks are used so that a sufficient volume of PZT material is present toconvert the desired amount of power from mechanical into electricalforms. As will be clear to those of skill in the art, an alternator maybe constructed with a different number of discs and/or discs with adifferent number of wedges. Also, the alternator designs disclosedthroughout this application may be substituted into any of thegenerators disclosed herein, or may be combined in various ways.

As the bimorphs bend, they generate an electric potential across theirfaces. The disks are electrically wired in parallel to build up thetotal current. An inverter/controller (not shown) converts this power tothe frequency and voltage needed by the electrical load. Theinverter/controller may also be responsible for regulating the fuel flowto a burner, if this is the desired source of heat, to match the powerneeds of the generator as the electrical load changes.

As will be clear to those of skill in the art, this embodiment of thegenerator 79 includes a pair of thermoacoustic engines 92 and 94arranged on opposites sides of the alternator 78. As the engines 92 and94 are preferably identical, only 92 will be described in more detail.It includes a nacelle 96 that is coaxial with the pressure vessel 80 anddefines an annular space between the outer surface of the nacelle andthe inner surface of the wall of the pressure vessel 80. A cold heatexchanger 98 and a hot heat exchanger 100 sandwich a regenerator 102inside the nacelle 96. As shown, a thermal buffer tube region is formedbetween the open end of the nacelle 96 and the hot heat exchanger 100 toprevent heat flow from the hot heat exchanger 100 to the gas in theannular space surrounding the nacelle 96. Also as shown, the nacelle 96may have an improved shape to improve the flow characteristics aroundthe nacelle.

The externally radiated noise from the raw core shown in FIGS. 4A and 4Bwould be an unacceptably high 105 dBA at full output power, due to theunbalanced design of the generator. The resonator body would vibratealong its axis with a 0.68 mm amplitude in response to the motion of thepiezoelectric disks. However, the sound level could be brought down toabout 70 dBA by placing a pair of generators end-to-end in a singlepressure vessel, operating out of phase with each other. Thisconfiguration would be similar to the one illustrated in FIG. 1. In thebalanced configuration, the radiated noise would have roughly equalcontributions from residual vibration and from the straining of theresonator wall due to the internal pressure swings. However, dependingon application, the main pressure vessel should additionally besurrounded with a second thin walled vessel, which would further isolatethe radiated noise and protect the inner vessel from damage. Forexample, for a 2 kW generator in the balanced configuration of FIG. 1(four 500 W engines with two 1 kW alternators), a 0.7 kg outer vesselmade of 0.32 mm (0.013″) thick steel could surround the inner vesselwith a 1 cm gap. The estimated externally radiated noise then comes downto a nearly inaudible 34 dBA.

Not shown in FIGS. 4A and 4B is an optional flow straightener or guidingvanes at the open end of the nacelle 96. As gas makes the sharp turnbetween annular and axial regions of the nacelle, net mean circulationsof the flow may occur due to flow separation at the lip of the nacelle.These steady currents may reduce the ability of the thermal buffer tubeto function as a thermal isolation layer. A screen or other perforatedstructure across the open end of the nacelle, and/or a vanes to helpguide the flow as it makes the turn, can serve to minimize thisdetrimental flow. These structures may also be chilled with the sameheat transfer fluid that runs through the cold heat exchanger, tofunction as secondary heat exchanger (see G. W. Swift et al., “AcousticRecovery of Lost Power in Pulse Tube Refrigerators,” J. Acoust. Soc.Am., 105 (2), Pt. 1, 711-724, 1999, the entire contents of which isincorporated herein by reference), to further define the temperatureprofiles in the working gas.

Also not shown in FIG. 4 is a means for stopping so-called Gedeonstreaming (see D. Gedeon, “DC Gas Flows in Stirling and Pulse-TubeCryocoolers,” in Cryocoolers 9, edited by R. G. Ross, Plenum, N.Y.,385-392, 1999, the entire contents of which is incorporated herein byreference), a net mean flow of mass due to the presence of sound, fromthe cold heat exchanger, through the regenerator, to the hot heatexchanger, and returning to the cold heat exchanger through the annularspace around the nacelle. This streaming flow is a parasitic load on theheat exchangers, robbing the engine of its efficiency. As taught by theLos Alamos group, the streaming can be stopped with an additionalflexible diaphragm (or light piston supported by a flexible bellows)that allows the passage of oscillating gas motions but not steady gasmotions, or with a so-called jet pump. Such a diaphragm may be placednear, but at a slight distance away from, the cold heat exchanger wherethe oscillatory motion of the diaphragm would be small, across thethermal buffer tube where it could also act to straighten the flow inthat region, or in the annular space between the nacelle and thepressure vessel.

As part of the design of a transducer to be used as an efficientalternator, it is important to not allow the reactive mechanicalimpedance of the transducer to be very much larger than the usefulnon-dissipative resistive component of the mechanical impedance that isused for generating electricity. (Mechanical impedance is the complexratio of the force to the velocity; reactance is the imaginary part andresistance is the real part.) Such excess reactance, if it cannot bebalanced by a canceling reactance of the opposite sign, takes away fromthe force available to generate electricity. Depending on its size andshape, the lowest natural resonance frequency of a PZT structure can bequite high, above several kilohertz-potentially much higher than theoperating frequency of the engines. Below its resonance, the PZT isbeing used in a frequency region where its stiffness controls itsoverall impedance. The higher the operating frequency can be raisedtoward the resonance frequency, the lower the stiffness reactance1(jωC_(M)) becomes, where j is the square root of −1, ω is the angularfrequency, and C_(M) is the mechanical compliance of the PZT structure.By way of contrast, electrodynamic transducers generally have arelatively low resonance frequency, in the few tens of hertz. Anelectrodynamic transducer is typically used above its resonancefrequency where it is mass controlled. The higher the operatingfrequency the larger and more troublesome the mass impedance jωM gets,where M is the moving mass. Therefore, piezoelectric transducers arefavored at high frequencies and electrodynamic transducers are favoredat low. For this reason high frequency sonar and ultrasonic transducerstend to be piezoelectric, while low frequency sonar and audiotransducers tend to be electrodynamic. For thermoacoustic applications,the crossover between the two types of transducers appears to be near300 Hz.

An important advantage in generating electricity piezoelectrically isthat the mechanical impedance of the piezoelectric material, because itis controlled by its stiffness rather than its mass, allows theoperating frequency of the generator to be increased resulting in analternator that is smaller in size and lighter in weight than acorresponding electrodynamic system operating at a lower frequency. Theavailable output power from a mass of PZT goes up proportionally withthe operating frequency (because the maximum energy that can beextracted per cycle is independent of frequency and there are morecycles per second at higher frequencies). In both alternatorembodiments, the part of the alternator that performs the energyconversion function and is the most massive, the PZT ceramic, isconfigured to have little motion. The moving mass is minimized bybringing mechanical power to the PZT through a relatively lightcoupling, such as the spokes of the first embodiment or the vertices ofthe bimorph wedges of the second embodiment, while allowing most of thePZT to stay relatively motionless. Providing for enough axial motion ofthe hub, which would be a difficult challenge at low frequencies usingpiezoelectric techniques, is easier at high operating frequencies wherethe acoustic displacements are smaller. With an electrodynamictransducer the moving coil or magnet is also part of the transductionprocess and so needs to be relatively massive to achieve high outputpowers. It is relatively easy to construct an electrodynamic transducerto have enough stroke to operate at a low frequency, but its largemoving mass makes operation at high frequencies difficult.

A second advantage in pushing toward higher operating frequencies isthat the length, and thus the volume, of a thermoacoustic engine goesdown proportionally with the acoustic wavelength, which scales inverselywith the frequency, with little penalty in output power. Engine weightand size is therefore also reduced. The ability to increase operatingfrequency of the engines appears to be limited by the performance ofheat exchangers at these shorter lengths, by the increase in parasiticthermal conduction as the hot and cold portions of the engine come intocloser proximity, and by the moving mass of the alternator/transducer.

In addition to making possible smaller alternator and engine sizes andweights, another advantage of piezoelectric alternators overelectrodynamic alternators is that the PZT alternator should be highlyefficient, above 95 percent efficiency—a modest improvement over the80-85 percent efficiency of a comparable electrodynamic alternator.

Advantages of the ring transducer over the bimorph transducer are: thereare fewer parts; the frequency of excitation of the PZT is twice that ofthe sound allowing for the use of half as much PZT; the PZT can be usedin the 3-3 mode allowing greater output power; the PZT is never intension where it is mechanically weak allowing greater output power; thevoltage induced during compression of the PZT is of the same sign as theoriginal poling voltage and thus does not have a tendency to de-pole theceramic which also allows greater output power; and it is difficult tooverdrive the transducer because the mechanical advantage in theconversion of axial hub force to radial spoke force decreases at higheramplitudes. Advantages that the two transducer types share are: theypresent a mass acoustic impedance which allows the resonator to be mademuch shorter; they have negligible internal thermal acoustic losses ontheir internal surfaces because there are no appreciable pressure swingsinside the transducer; there are negligible internal viscous shearlosses because no fluid is made to pass through any structures; and theyhave a small volume because they do not use a “back-volume” common tomany other transducer types.

Since there are no sliding seals or lubrication in either embodiment ofthe thermoacoustic generator, there does not appear to be anything towear out or require maintenance, as long as all parts are designed forinfinite fatigue life. Depending on the application, however, in acomplete system it may be necessary to include a small water pump and/orfan for the ambient cooling loop, which may require some minimalmaintenance. Furthermore, the generator is environmentally friendlybecause the working fluid is a benign gas such as helium, helium-argonmixtures, helium-xenon mixtures, or air—no environmentally harmfulsubstances are used.

Thermoacoustic engines can use a wide variety of heat sources andtemperatures, and are adaptable to many applications. For example, in adevice with multiple thermoacoustic stages, as was illustrated in FIG.1, it is quite straightforward to convert some of those stages fromengines into chillers or refrigerators, making a power generator thatalso delivers chilled water. Referring to FIG. 1, with the verticalorientation shown in the figure, it is preferred that engines 26 and 30be converted into refrigerators while engines 24 and 28 remain heatdriven engines so that gas in the thermal buffer tubes of all theengines and refrigerators are gravitationally stable, with either hotheat exchangers on the top or cold heat exchangers on the bottom of thethermal buffer tubes. Such an arrangement is illustrated in FIG. 23.

The coaxial implementation of the thermoacoustic-Stirling engine shownherein is very compact, which results in weight and volume savings and,we believe, leads to lower acoustic losses than the original Los Alamosengine. The heat exchanger pressure vessel penetrations and thearrangement of the engine components suspended within the pressurevessel are chosen to minimize the adverse thermal expansion stressesassociated with the hot heat exchanger temperature. By having the hotpenetrations on one side of the nacelle and the cold penetration on theother side, as can be seen in FIGS. 4A and 4B, thermal expansionstresses are relieved by allowing the nacelle to tip slightly as the hotcomponents slightly expand. Furthermore, because the nacelle is not partof the pressure vessel, it can be a thin wall hollow structure, whichminimizes losses due to thermal conduction, both along the nacellebetween the hot and cold heat exchangers, or through the nacelle betweenthe hot heat exchanger and the gas in the annular passage outside thenacelle.

EXAMPLE 3 Ring Alternator with Spokes and Azimuthal Springs

Azimuthal springs may be added to the embodiment of the alternator inFIGS. 2A and 2B, as shown in FIG. 5, to lower the ring resonantfrequency in the generator to twice the acoustic frequency. Thisembodiment shares with the embodiment of FIGS. 2A and 2B two forceamplifying mechanisms for matching the mechanical impedance of thetransducer to that of the engine: the conversion of axial force on thehub 104 into radial tension on the spokes 105, and the conversion oftension in the spokes to compression of the ring 106.

The alternator 103 of FIG. 5 may be used in place of the alternator ofFIGS. 2A and 2B. As with the alternator of FIGS. 2A and 2B, thealternator 103 includes a ring 106 of piezoelectric material that formsa perimeter member. The ring includes stacks of piezoelectric materialextending between supports 108. Azimuthal springs 107 are placed betweenthe stack of piezoelectric material and each support 108. This placesthe springs in series with the piezoelectric material. The supports 108are in turn interconnected with the wall of the pressure vessel byspring supports 109. The hub 104 is interconnected with supports 108 byspokes 105. As with the embodiment of FIGS. 2A and 2B, there arepreferably several layers of spokes that are generally parallel to eachother. The hub 104 is interconnected with an alternator face ordiaphragm, as was described with respect to FIG. 2B.

The azimuthal springs 107 distributed within the piezoelectric ring 106constitute an additional mechanism of force amplification. Thealternating component of the input force to the ring is much reducedwhen the ring is driven at resonance. The natural frequency of a 5″diameter PZT ring without springs is about 8 kHz. In a rigid ringwithout springs, as in the embodiment of FIGS. 2A and 2B, most of theforce applied to the ring works against the stiffness of the ring whendriving the ring below its resonance frequency. The addition of thesprings 107, as well as the possible addition of mass to the supports108, can be used to lower the resonant frequency of the ring to twicethe operating frequency, which in the calculations performed so far hasbeen taken to be 800 Hz for the radial motion of the ring and 400 Hz forthe sound.

Resonating the ring leads to several advantages: First, the alternatingmembrane tensile forces in the spokes are greatly reduced—byapproximately a factor of 50, determined by the range of tuning mismatchbetween the engine and the ring as a function of operatingtemperatures—although the mean component of the membrane tensile forceremains the same. The peak force in the spokes, therefore, is reduced byapproximately a factor of two. With the reduction in the peak spokeforce, the spokes and diaphragm can be made about a factor of twolighter, allowing a higher operating frequency. Secondly, the spoketension is nearly constant which diminishes mechanical fatigue,especially away from the spoke ends. Third, a constraint of a “bestsized” alternator of 20 kW that is present without the springs anddescribed below in Example 4 is removed so that alternators of smallerpower are made possible.

EXAMPLE 4 Summary of Piezoelectric Ring Alternator Calculations

In order to teach one skilled in the art how to construct apiezoelectric ring alternator without undue experimentation, anelectromechanical equivalent circuit for the quadratically drivenpiezoelectric ring alternator with spokes will be developed here. It isa conventional linear model that uses some linear and nonlinearproperties of a so-called “hard” PZT piezoelectric ceramic (Navy typeIII) as parameters, as an illustrative piezoelectric material. The modelwill be used to estimate the load resistance that maximizes thealternator efficiency and that maximizes the output power. To furtherteach the use of the ring alternator configuration, an example designcalculation will be illustrated here. The analysis will show that thealternator efficiency should be about 98 percent and the output powerdensity should approach 20 W/cm³ of PZT (at 400 Hz) with an appropriatechoice of load resistance. The model includes azimuthal springs betweenthe PZT stack and its support, but also holds when azimuthal springs areabsent. The best natural size of the generator when azimuthal springsare not used will also be illustrated.

Consider an N_(S) sided polygonal ring alternator of N_(S) stacks of PZTand N_(S) sets of spokes. One of N_(S) equivalent slices of thealternator without azimuthal springs, centered on one set of spokes, isshown in FIG. 6A. FIG. 6B shows an equivalent slice of the alternatorwith azimuthal springs. The same force and dimension labels are used ineach. Each stack of PZT has length l, cross sectional area A_(C), and ismade up of n layers of PZT each t thick, such that l=nt. The spokelength taken from the center of the alternator to the centerline of thePZT stacks is a. Half the angle subtended by two sets of spokes isπ/N_(S) (radians). Although the supports and springs should ideally takelittle space, a sin(π/N_(S)) is not assumed to be equal to l/2 toaccount for the room taken up by them. The radial velocity of thesupports is u_(R), and the force of one set of spokes on its support isF_(S), both taken as positive inward. The azimuthal springs have acombined compliance C_(A) per segment along their axis, in the directionof the line connecting the support and the PZT stack (the two springsshown in FIG. 6B each have compliance C_(A)/2). The azimuthal springsare assumed to be infinitely stiff in the transverse direction so thatthe PZT stack is constrained to move in the left-right direction of FIG.6B with the same velocity u_(R) as the supports.

PZT is conventionally described by a coordinate system where the 3direction is the poled direction. (In its manufacture, metal electrodesare fired onto two surfaces of PZT. A high voltage applied across theelectrodes while at high temperature polarizes the ceramic in the 3direction, perpendicular to the electrode surfaces, making the ceramicpiezoelectric.) The other two directions perpendicular to 3 haveequivalent properties and are called 1 and 2. PZT elements are stackedtogether in the alternator, electrodes touching, with the 3 direction ofeach element alternating in direction. Every other electrode is wiredtogether. The PZT elements are mechanically in series but are allelectrically in parallel.

The direction of the components of stress T, strain S, electric field E,and electric displacement D in the PZT elements are denoted withsubscripts. The elements of the stack are mechanically free on theirsides, so T₁=T₂=0. By symmetry, E₁=E₂=0 and D₁=D₂=0. The transversestrains S₁ and S₂ are nonzero but are not interesting. This simplifiesthe relations between the remaining PZT element variables:S ₃ =s ₃₃ ^(E) T ₃ +d ₃₃ E ₃,  (1)D ₃ =d ₃₃ T ₃+ε₃₃ ^(T) E ₃,  (2)where anisotropic material properties of the PZT are given by theseparameters: s₃₃ ^(E) is a component of the compliance matrix at constantelectric field (signified by the superscript E), Δ₃₃ ^(T) is a componentof the dielectric constant matrix at constant stress (signified by thesuperscript T), and d₃₃ is a piezoelectric coupling constant. Fiveadditional relations are needed to connect the piezoelectric elements tothe macroscopic inputs and outputs of the alternator. The extensionΔξ_(A) of the spring is related to the force applied to it byT ₃ A _(C)=Δξ_(A) ^(˜) C _(A).  (3)The combined extension of the spring and PZT stack are related to theradial support motion by(Δξ_(A) +S ₃ l)/2=−u _(R) sin(π/N _(S))/(jω),  (4)where j is the square root of −1 and ω is the angular frequency 2πf ofthe ring motion (at twice the acoustic frequency). Newton's second lawapplied in the radial direction to the alternator slice shown in FIG. 6BisF _(S)+2A _(C) T ₃ sin(π/N _(S))=jω(ρ_(PZT) A _(C) l+m)u _(R),  (5)where ρ_(PZT) is the PZT density and m is the mass of the support andits two attached springs. The output current I (taken as positiveinward) and voltage V of the alternator are related to piezoelectricvariables byI=jωnN _(S) A _(C) D ₃,  (6)V=tE ₃.  (7)The seven equations (1)-(7) are the starting point of the model of, sofar, nine variables T₃, S₃, E₃, D₃, Δξ_(A), F_(S), u_(R), I, and V. Anadditional relation can be made to specify the electrical load; forexample, a simple load resistor R_(L)I=−V/R _(L)  (8)

Substituting (1) for T₃ in (5), using (3), (4) and (7) to eliminate S₃and E₃, multiplying through by N_(S), and defining F_(R)=N_(S) F_(S) asan arithmetic (not vector) sum of the radial spoke forces F_(S), givesan impedance form of Newton's law:[jωM+1/(jωC _(M)(1+K))]u _(R) +NV/(1+K)−F _(R)=0,  (9)where the ring mass is given byM=(ρ_(PZT) A _(C) l+m)N _(S),  (10)the ring compliance is given by $\begin{matrix}{{C_{M} = {\frac{{as}_{33}^{E}}{2{\pi A}_{C}}\frac{1}{\left( {N_{S}/\pi} \right){\sin\left( {\pi/N_{S}} \right)}}\frac{l}{2{{a\sin}\left( {\pi/N_{S}} \right)}}}},} & (11)\end{matrix}$the piezoelectric transduction process is described by the coefficient N$\begin{matrix}{{N = {\frac{{nN}_{S}A_{C}d_{33}}{{as}_{33}^{E}}\frac{2{a\sin}\left( {\pi/N_{S}} \right)}{l}}},} & (12)\end{matrix}$and a factor, 1+K, appears because of the presence of the azimuthalsprings, where K is the ratio of the spring compliance to PZT complianceK=C _(A) A _(C) /ls ₃₃ ^(E).  (13)If azimuthal springs are not present, K should be set equal to zero. Thelast factors in (11) and (12) are nearly unity. They are a smallcorrection for the finite size of the support and springs. The next tothe last factor in (11) is also nearly unity. It is a correction to theround ring result for a polygon of a finite number of segments N_(S).

The mechanical resonance frequency of the ring is found by setting themechanical impedance jωM+1/[ωC_(M)(1+K)] of (9) equal to zero:ω₀ ²=1/[MC _(M)(1+K)].  (14)The forces in the spokes will be minimized at this frequency. Thenaturally high resonance frequency of a PZT ring can be brought down totwice the acoustic frequency by increasing K and/or M.

Kirchhoff's law for the electrical side of the alternator comes fromsubstituting (2) into (6), using (1), (3), (4) and (7) to eliminate T₃,S₃, E₃, and Δξ_(A), and using (8) to eliminate I:−Nu _(R)/(1+K)+(jωC ₀+1/R _(L))V,  (15)where the blocked (u_(R)=0) output capacitance C₀ of the alternator isgiven byC ₀=[1−k ₃₃ ²/(1+K)]nN _(S) A _(C)ε₃₃ ^(T) /t,  (16)and the electromechanical coupling constant k₃₃ is given by k₃₃ ²=d₃₃²/s₃₃ ^(E)ε₃₃ ^(T).

Newton and Kirchhoff's laws, (9) and (15), can be represented by theequivalent circuit shown in FIG. 7A, which obeys the same mathematics asEqs. (1)-(16). An ideal transformer with turns ratio N, which has unitsof N/V, represents the electromechanical transduction terms in (9) and(15). The compliance of all the azimuthal springs is represented by ashunting capacitor C_(B)=K C_(M). Following common practice, theresistors R_(M) and R₀ are added to represent mechanical and electricaldissipation within the PZT. The manufacturer of the PZT specifies amechanical Q_(M) from which to derive the mechanical loss resistanceR_(M),R _(M)=1/ωC _(M) Q _(M),  (17)and a dielectric loss tangent tan(δ) which is used to calculate R₀,R ₀=1/ωC′₀ tan(δ)=t/ωnN _(S) A _(C)ε₃₃ ^(T) tan(δ)  (18)where C′₀ is C₀ without the so-called “motional” factor [1−k₃₃ ²/(1+K)].Resistors R_(J) and R_(B) are included to represent mechanical lossesoutside the PZT, in such places as adhesive attachments of the PZT or inthe azimuthal springs, respectively, but for now both are assumed to bezero. Their true value should be determined experimentally.

Maximizing the efficiency of the generator requires the minimization orcancellation of the total reactance of the alternator. For example, apower factor inductor may be added across the alternator output tocancel the current through C₀. Part of the design task for the generatoras a whole is to balance the stiffness reactances (which store potentialenergy) of C_(M) and C_(B), the spoke stiffness and the gas stiffness ofthe thermoacoustic engines with the inertial reactances (which storekinetic energy) of the ring, hub, diaphragm and the spoke masses.Assuming all these reactances balance out, and R_(J) and R_(B) areindeed small enough that they can be neglected, the equivalent circuitcan be reduced down to the circuit of FIG. 7B. The useful output poweris Π_(o)=|V|²/2R_(L), where |V| is the output voltage peak amplitude.The input power is Π_(i)=|V_(i)|²/2R₁, where the input resistance R_(i)is the series-parallel combination R_(i)=R_(m)+(1/R_(L)+1/R₀)⁻¹, andR_(m)=R_(M)/N² and |V_(i)|≦F_(R)/N are the mechanical resistance andinput force reflected over to the electrical side of the transformer.The efficiency e is Π₀/Π_(i), which after some manipulation is to goodapproximatione=R _(L) R ₀/[(R _(L) +R _(m))(R _(L) +R ₀)]  (19)For an alternator to have a reasonable efficiency, R_(m) will have asmall value because it is a series element and R₀ will have a largevalue because it is a parallel element. Combining (11), (12), (17) and(18) with the definition of R_(m) gives for the ratio, which should belarge,R ₀ /R _(m) =k ₃₃ ² Q _(M)/tan(δ).  (20)Notice that this expression depends only on the material properties ofthe PZT rather than on any other design choices. Of the commoncommercially available versions of PZT, Navy type III, also known asPZT-8, is very likely the best choice for this alternator because of itshigh Q_(M) and low tan(δ), even at high amplitudes. It was developed forhigh power sonar projectors. Typical values for PZT-8 from onemanufacturer are k₃₃ ²=0.44, Q_(M)=1050 and tan(δ)=0.004 at lowamplitudes. At high amplitude (E₃=4 kV/cm rms), tan(δ) degrades to 0.01and Q_(M) drops to around 350. For these degraded high amplitude valuesR₀/R_(m) is still high at 15,000. The efficiency given by (19) isplotted in FIG. 8 for PZT-8 at high amplitude on a log-log plot. Theefficiency is greater than 90 percent over a two order of magnituderange of R_(L). Because R₀/R_(m) is so high there is a comfortably broadrange of R_(L) that will give high efficiency. The curve is symmetricabout the geometric mean of R_(m) and R₀. The highest efficiencytherefore occurs withR _(L MAX EFFICIENCY)=(R ₀ R _(m))^(1/2)=(R ₀ R _(M))^(1/2) /N.  (21)The value of efficiency e for this value of R_(L) is 98.4 percent. Apiezoelectric alternator can potentially have such a high efficiencybecause with PZT-8 Q_(M) is very high and tan(δ) is very low, as long asother mechanical loss mechanisms represented by R_(J) and R_(B) in theequivalent circuit are kept small.

The load resistance can also be chosen to maximize output power ratherthan efficiency. The PZT has a maximum electric field E_(3 MAX) and amaximum stress T_(3 MAX) that it can tolerate without damage. Usingthese values for E₃ and T₃ in R_(L)=−V/I, (2), (6) and (7), and assumingthat the reactive currents are negligibly small, an approximateexpression for the value of R_(L) that maximizes the output power isR _(L MAX POWER) =E _(3 MAX) t/T _(3 MAX) ωnN _(S) A _(C) d ₃₃.  (22)As it fortuitously turns out, these two optimized values for R_(L) arenearly the same as can be seen by taking their ratio: $\begin{matrix}{{\frac{R_{L\quad{MAX}\quad{POWER}}}{R_{L\quad{MAX}\quad{EFFICIENCY}}} = \left( \frac{E_{{3\quad{MAX}}\quad}^{2}ɛ_{33}^{T}{\tan(\delta)}}{T_{{3\quad{MAX}}\quad}^{2}{s_{33}^{E}/Q_{M}}} \right)^{1/2}},} & (23)\end{matrix}$which has the value of 0.87 for PZT-8. Like (18), this expression isindependent of design decisions; it depends only on material properties.It should therefore be quite straightforward to simultaneously have highefficiency and high power from the alternator.

The maximum power that can be taken from the alternator per unit volumeof PZT, π_(MAX), is given by (E_(3 MAX) t)²/(2 R_(L MAX POWER) N_(S) lA_(C)) which reduces to a maximum power density in the PZT ofπ_(MAX) =ωd ₃₃ E _(3 MAX) T _(3 MAX)/2.  (24)

For PZT-8 limiting values for E_(3 MAX) and T_(3 MAX) are 8 kV/cm and7000 PSI, respectively, when no mechanical or electrical bias ispresent, i.e. when the electric field and mechanical stress swingsequally in the positive and negative directions. This condition leads toa power density of 23 W/cm³ for an example acoustic frequency of 400 Hzand ring motion frequency of 800 Hz. The negative swings of the electricfield and peak mechanical stress in tension are what limit this powerdensity. The ceramic is weak in tension, and both negative voltages andtensile stresses tend to de-pole the ceramic. The quadratically drivenring alternator (with spokes) has the advantage, however, that the PZTis always placed in compression and the voltages induced are positive—inthe same direction as the original poling voltage. Thus it may bepossible to drive the PZT to higher levels. A survey of the literaturesuggests that E_(3 MAX) and T_(3 MAX) may be pushed to as high as 15kV/cm and 12,000 PSI, respectively, when both are only applied in theiradvantageous directions. If this is so, then π_(MAX) may be as high as75 W/cm³, which is very high. With the more conservative 23 W/cm³ powerdensity, the thermal power to be dissipated would be 0.37 W/cm³ at 98.4percent efficiency, but at 75 W/cm³ the thermal dissipation would growto 1.2 W/cm³. At some point the PZT could be thermally limited. Thetolerance of higher PZT power densities should provide a safety factorfor short mechanical overloads, however.

Azimuthal springs give some flexibility in the design by allowingadjustment of the overall radial motion u_(R) of the ring, which in turnallows adjustment of the hub motion to best match the acoustic motiondelivered by the engine for a given diameter of the ring. If a choicehas been made not to use azimuthal springs, however, then there is abest natural diameter and power for the generator because of the need tomatch the alternators to the engines. The acoustic volume displacementdetermines the necessary volume swept out by the faces of thealternator, which in turn is related to the PZT strain through the spokegeometry. The alternators and engines are then matched by adjusting theamount of PZT in the alternators so that the maximum amount of powerabsorbed by the alternators and converted into electricity is the sameas the maximum power produced by the engines. An example calculation ofmatching an alternator without azimuthal springs to a thermoacousticengine is shown next.

We will require that the alternators and the engines fit in a pressurevessel with the same internal radius R for simplicity of the vessel. Theoutput power of the engines will be scaled with R² to keep the acousticpower density, gas velocity u_(A) and pressure amplitude at thealternator fixed at the values 7.18 W/cm², 18.2 m/s and 88.6 kPa,respectively, of a thermoacoustic-Stirling engine numerical model forthe design of FIG. 1. For an effective moving area of the diaphragmassumed to be 80 percent of πR², the peak displacement of the diaphragmζ is then given by ζ=u_(A)/(0.80) ω_(A)=9.0 mm, where ω_(A) is theangular frequency of the sound (=2π400 Hz). The spokes bend in acomplicated manor, but for simplicity they will be assumed to have theirflexing confined to a short region near their ends (pivoting like ahinge) and an effective length h=K_(h) a, where K_(h), will be taken tobe 0.90. A fatigue analysis estimates that the safe peak tensile stressT_(S) in spring steel spokes is 30 kPSI. The spokes will have strainS_(s)=T_(S)/Y_(S)=30 kPSI/30 MPSI=0.001 while under tension, where Y_(S)is Young's modulus of the spokes. The maximum PZT strain S₃ isapproximately T₃ s₃₃ ^(E)=6.8×10⁻⁴. Using the Pythagorean theorem, theinward radial motion S₃a of the PZT supports can be related to ζ byS₃a=h−h [(1+S_(s))²−(ζ/h)²]^(−1/2). Solving for a gives the alternatorradius best matched to the engines:a _(BEST)=ζ[2K _(h)(S ₃ +K _(h) S _(S))]^(1/2)=16.9 cm,  (25)after plugging in values. In order to leave room for the PZT and supportsprings, the pressure vessel diameter should be 2R≅2a/0.80=42.3cm=16.6″. This then leads to the power from each thermoacoustic-Stirlingengine Π_(A)=π_(A)πR²=10.0 kW. As each engine is about 6″ in length, agenerator in the balanced configuration of FIG. 1, with four 10 kWengines and two 20 kW alternators, would generate 40 kW (54 horsepower)of electricity in a pressure vessel about 34″ long and 17″ in diameter.

The alternator design is pretty much determined at this point in theanalysis. The number of sections N_(S) can be picked for convenience tobe 6, say. Also for convenience, select a≅l=nt=16.0 cm, n=16 and t=1.0cm. With π_(MAX)=23 W/cm³ and the requirement for 20 kW gives A_(C)=9.06cm², which if the PZT is chosen to be in the form of stacked rounddisks, would give the disks a diameter of 3.40 cm diameter. The totalmass M of PZT needed for each alternator is only 7.55 kg. The alternatorcan be designed for maximum output power rather than efficiency sincethe efficiency peak is so broad. At full drive levels, with ζ=9.0 mm,the alternator output voltage is 5875 V_(RMS), the output current is3.40 A_(RMS), the maximum PZT stress T₃ is 8200 PSI (a bit higher thanthe targeted 7000 PSI) and maximum field E₃ is 8.3 kV/cm (also, a littlehigher than 8.0 kV/cm). The alternator efficiency (acoustic toelectrical) e is 98.2 percent, assuming no additional losses. Ifdesired, n can be varied to trade output voltage for current, as if itcontrolled the turns ratio of an output transformer. Other variables ofinterest, derived from the equations above are: R_(L) 1726 Ω C_(M) 4.15× 10⁻¹¹ m/N R_(M) 1370 Ns/m N 9.25 N/V C₀ 45.6 nF R₀ 246 kΩ R₁ =R_(M)/N² 16.0 Ω f₀ = ω₀/2π 2843 Hz |F_(R)| 287 kN |u_(R)| 0.563 m/sphase angle by which u_(R) leads F_(R) 75.4° phase angle by which u_(R)leads V 21.6° power dissipated in R_(M), |u_(R)|² R_(M)/2 217 W powerdissipated in R₀, V_(RMS) ²/R₀ 140 W

The alternator efficiency of 98.2 percent (acoustic to electric powerconversion) coupled with a projected engine thermal efficiency of 33percent (thermal to acoustic conversion) gives a thermal to electricalconversion efficiency for the generator of 32 percent. The generatoralso appears to be potentially fairly compact and quite lightweight.

Azimuthal Spring Embodiment

The desired spring constant and range of motion of the azimuthal springsfor use in the piezoelectric ring of FIG. 5 tends toward that of conicalBellville washers, although the springs must be designed in such a waythat they do not suffer fretting wear or fretting fatigue at theirpoints of contact with the supports or the piezoelectric material. Apreferred embodiment of the azimuthal springs is shown in FIG. 16. Thefigure is a sectional diagram of a solid of rotation about the centerline. The spring 270 consists of a first piece 272 and a second piece274 that are attached at their contact by a means such as welding,brazing or soldering, as shown at 276. The main flexible energy-storageelement is a pair of Belleville-like cone spring portions 278 and 280which are part of the first and second pieces 272 and 274. The cones 278and 280 attach to a base plate 282 and 284, respectively, on either endof the spring through a central pedestal 286 and 288, respectively.Fretting is avoided by use of the pedestal and the joining of the cones.As will be clear to those of skill in the art, other spring designs andapproaches may be used in place of the bevel-like design disclosedherein.

As an alternative to the bevel-like cone springs, an elastomericmaterial may be used. For example, a simple block of low mechanical-lostelastomeric material may be positioned between each stack ofpiezoelectric material and the corresponding keystone-shaped support.

EXAMPLE 5 Performance of the Bimorph Generator

High efficiency and low radiated noise appear to be key advantages of athermoacoustic-Stirling piezoelectric generator. Other advantages overother technologies seem to be a smaller size and weight. For the designof FIGS. 4A and 4B, similar to the calculation above, the generator coreefficiency (heat into the hot exchanger to electricity out) has beenestimated to be 32 percent. This is the product of a 34 percent engineefficiency (heat to sound) and a 95 percent alternator efficiency (soundto electricity). The engine efficiency is 51 percent of the ideal Carnotefficiency. The total mass of the core components shown in FIGS. 4A and4B is 12.8 kg (28 lb) (13.5 kg with a noise proofing shell), most ofwhich (9.7 kg) is in the PZT alternator. The volume is 7.8 liters. Witha 2.0 kW output this gives specific output powers of 260 kW/m³ and 156W/kg for the thermoacoustic generator core.

This core efficiency of 32 percent might be the appropriate systemefficiency when the generator is embedded within a large system. Forexample, for power generation on a naval ship, heat could come from gasturbine waste heat, while the heat rejected from the cold exchangerscould be taken away by a ship cooling loop, and the output power couldbe used directly or used after it was rectified to DC power. However, inother applications additional parts may be needed, which would degradethese numbers. For example, a stand-alone mobile power generator wouldrequire a burner. To maintain a high system efficiency the combustionair should be pre-heated with the exhaust from the hot heat exchangerthrough a recuperator. Not all of the heat of combustion will be able tobe captured in this way, however, so there will be a burner efficiency(heat value of fuel to heat into the hot exchanger) that is less thanone. An estimate might put this at 85 percent, bringing the systemefficiency down to 27 percent. The burner and recuperator also add tothe mass and volume of the system. For the same mobile application, ameans of rejecting the heat from the cold heat exchangers must also beprovided. In some cases this may be done passively with heat pipes andnatural convection cooling. However, it may be necessary to use aconventional cooling loop consisting of a water/glycol mixture,plumbing, a pump, an external heat exchanger and a fan. Allowing 100Wfor the pump and fan brings the system efficiency down from 27 percentto 26 percent. Finally, the high frequency output of the PZT should, inmost cases, be rectified to DC and then converted into 120/240V 60 Hz,as needed, with a power inverter, which may have an efficiency (highfrequency power in to 60 Hz power out) of 95 percent, bringing theoverall system efficiency down to 25 percent. A fuel tank and overallenclosure or frame would be needed as well, which, with the otherauxiliary parts, degrades the specific power density numbers.

Nevertheless, if we allow for an additional 7.5 kg of mass and 19 litersof volume (which includes 11 liters for a 3.0 gallon fuel tank), and usethis as an estimate for a design including all these auxiliarycomponents to make the system self contained and mobile, athermoacoustic generator still looks good compared to the bestcomparable generator from Yamaha Corporation, the model YG2800i,especially on efficiency and radiated noise level where our estimatesare the most reliable: Thermoacoustic Generator Estimate Yamaha YG2800iOutput power 2000 W 2500 W Weight 21 kg 30 kg Volume 27 liter 81 literPower per weight 95 W/kg 84 W/kg Power per volume 74 kW/m³ 31 kW/m³ W-hrof electricity per gallon 8920 5375 of gasoline Efficiency (fuel energyto 25% 15% electricity) Full power radiated noise level 34 dBA 67 dBAOil to change no yes

EXAMPLE 6 Other Embodiments

FIGS. 2-4 show a corrugated diaphragm as a means for sealing thealternator piston faces to the pressure vessel while allowing axialmotion of the faces over a range of travel. Alternative means for doingthe same are shown in FIGS. 9, 10 and 11. In each of the three figures,a cross-sectional view of an alternator, along with a portion of thepressure vessel is shown. Referring first to FIG. 9, the pressure vessel110 surrounds the PZT ring 112, which is attached to the pressure vesselwall 110 by support springs 114. Spokes 116 interconnect the ring 112with the hub 118. The alternator has a pair of alternator faces 120 and122. In FIG. 9, the hub 118 and faces 120 and 122 are shown displaced tothe left. Roll socks 124 and 126 interconnect and seal faces 120 and 122to the pressure vessel wall 110. The roll socks are preferably in ringsof a half turn of flexible material, to provide a flexible seal.

FIG. 10 shows another embodiment with a similar pressure vessel wall 128surrounding a ring 130, with the ring interconnected with the wall byspring supports 132. Spokes 134 interconnect the ring 130 with the hub136. Alternator faces 138 and 140 are connected to the hub 136. As withFIG. 9, the hub 136 and spokes 134 are shown displaced to the left. Inthis embodiment, bellows 142 and 144 connect the faces 138 and 140,respectively, to the pressure vessel wall 128. This is another approachto providing a flexible seal.

FIG. 11 shows another alternative version, using a clearance seal. Aportion of the pressure vessel wall is shown at 146, the ring is shownat 148 and the support interconnecting the ring 148 with the wall 146 isshown at 150. The spokes 152 interconnect the ring 148 with the hub 154,and are shown deflected to the left. Alternator faces are shown at 156and 158. The face 156 may be said to have a piston skirt 160 which fitsclosely into a precision cylinder or housing 162, so as to provide avery small gap 164. The gap 164 should be small enough that negligiblegas is able to pass through the gap in response to motion of the faces,but not so small as to cause undue losses due to viscous shear of thegas in the gap. A similar clearance seal is provided for face 158.

FIG. 12 shows an alternative version of the ring transducers of FIGS.2A, 2B and 5. In this embodiment, the piezoelectric ceramic 200 is inthe form of a striped cylinder. It is a single piece of ceramic withstriped electrodes 205 painted around its perimeter. Alternateelectrodes are connected together to put the segments of the ringelectrically in parallel. As an alternative to flat shim spokes, FIG. 12shows wire spokes 201, made, for example, from high strength steel musicwire. Also, as the above calculation for the ring alternator showed,there is a best diameter, and therefore a best output power, for thering alternator that depends on the mechanical compliance of both thepiezoelectric ring and the spokes. FIG. 12 shows a technique forconstructing a lower power alternator that is matched to lower powerengines: increasing the compliance of the spokes. In this case, eachspoke is coiled into a spring 206 near the ends of the spoke. Placingthe coiled part near the piezoelectric ring rather than near the hub 204lowers the moving mass of the alternator, allowing a higher operatingfrequency. The spokes are attached, for example by brazing, to supportbars 202. In one embodiment, the support bar 202 is separated from thepiezoelectric ring 200 by an insulator 203. However, if the ring 200 ispatterned with electrodes 205 such that electrodes on either side of thesupport bar 203 are at the same potential, and no other electrodes arepatterned under the support bar, then the insulator 203 becomesunnecessary. The support bar, piezoelectric ceramic ring, and insulator(if used) may be joined with adhesive.

The techniques of FIGS. 12, 2A, 2B and 5 may be combined in variouscombinations. FIG. 13, for example, shows wire spokes 224, with coils222, attached to a keystone-like support 221, with a polygonalpiezoelectric ceramic ring made up of straight stacks 220 of ceramicelements. As another alternative, azimuthal springs may be included inany embodiment herein.

FIG. 14 shows a detail of an alternate means for attaching spokes 247 toa striped ring piezoelectric ceramic 244. This figure also shows analternate means for decreasing the size and power of a ring transducer:adding compliance to the spoke support structure. The spokes 247 areattached to a yoke 243, which is attached to a clamp 241 via elasticmembers 240 formed in the clamp by cutting a rounded bottom notch 248 inthe clamp. An insulator 242 may or may not be needed, depending on howelectrodes 245 are patterned on the ceramic ring and how they are wiredtogether with leads 246. If the region under all of the clamps 241 ofthe alternator are held at the same potential (such as ground), then theinsulators 242 may be omitted. The design of FIG. 14 has the advantageover that of FIG. 12 in that no adhesive is placed under tension, whereit is weak. Some adhesive between the clamp 241 and/or the insulator 242and ring 244 may be desirable to avoid motion of the clamp due to therelatively weak axial forces from the spokes when they are flexed.

FIG. 15 shows a detail of another alternate means for attaching spokes263, this time chosen to be constructed of flat shim stock. Multiplespokes 263 are joined to spacers 264 and mounting blocks 265, forexample by furnace brazing. A yoke and clamp piece 260 attaches to themounting blocks 265, for example with screws. The legs of yoke 261 maybe made elastic, if desired, by the cutting of multiple round bottomnotches 262, so as to allow the construction of a lower power alternatoras described earlier.

Spoke End Preferred Embodiment

The spokes for the embodiments discussed therein may be fabricated fromstacks of thin uniform thickness plates of strong fatigue resistantmaterial, such as spring steel. The spokes, keystone supports and hubmay be all cut out of different regions of the same plate. Many platesare stacked on top of one another, separated by shim plates in thesupport and hub regions. The whole assembly is joined into one unit by aprocess such as oven brazing. The spoke-support-hub assembly then formsa compliant mechanism, with solid supports and hub, and spokes which arefree to flex in the axial direction.

With axial motion of the hub, for spokes that are not too thick, thebending of the spokes is confined to a small region near the spoke endsnear the supports and the hub. The spoke ends are exposed both tomembrane stress due to radial tension in the spokes, and bending stress.The spokes can be treated as cantilever beams under simultaneous bendingand tension in a conventional manner for the selection of theirdimensions. In such a case, the bending stress, which is proportional tothe curvature of the spoke, is a function of the tension in the spoke aswell as the thickness and axial displacement of the spoke. The higherthe membrane tension, the sharper the curvature of the bend of the spokeend, and the higher the bending stress. However, for a constant tensilemembrane stress in the center of the spoke where the curvature isnegligible, it will be found that the curvature of the spoke end willincrease with decreasing spoke thickness in such a way that the ratio ofthe bending stress at the spoke base to the tensile stress at the centerof the spoke is independent of the thickness. Thus, it is not possibleto make the bending stress negligible by decreasing the thickness of thespokes.

For a straight spoke 290, shown in FIG. 17, the tensile force carryingcapacity of the spoke will be limited by the bending stress at the spokeend. Most of the rest of the spoke will not be fully stressed, and thespoke will be unnecessarily massive. The excess mass will limit thehighest operating frequency of the generator, but this might be anacceptable trade-off, in some cases, for the simplicity of the design.

A tapered spoke end 292, as shown in FIG. 18, is the preferred, higherperformance, embodiment for a spoke. It has a nearly triangular region294 near the base 296 (toward either the hub or the support) wherebending stress dominates the total stress, a straight region 297 awayfrom the base where membrane tensile stress dominates, and atransitional region 298 in between where both membrane and bendingstress is important. The shape of the spoke is optimized so that themaximum total stress at any radial position along the spoke, which isthe sum of the bending and membrane tensile stresses, is a constant.Thus, each portion of the spoke is maximally stressed and the spoke hasthe minimum mass. The ratio of the width of the spoke base to the widthof the straight section of the spoke is approximately proportional tothe square of the angle of deflection of the spoke. Thus, the optimalshape will depend on the diameter of the generator and the displacementof the hub.

An alternate embodiment of the spoke-support-hub structure is to stackstraight strips of spring steel in the shape of a cross—first ahorizontal strip, then vertical, then horizontal and so on. At the fourends of the cross are placed trapezoidal strips made of material of thesame thickness material as the spokes, in the gaps between the spokes,to form the keystone-shaped supports. The whole assembly may be joinedtogether by braze, adhesive, or fasteners, although fretting fatiguewhere the laminations cross at the spoke ends may be a problem whenusing fasteners. This four section spoke-support-hub structure does nothave optimally low mass, but it can be constructed at low cost.

Resonant Ring without Spokes Embodiment

FIG. 19 shows another embodiment of a piezoelectric alternator 300 thatcan be used as an acoustic or resonant mass. It is based on a resonantring 302 without spokes. The ring 302 is composed of a plurality ofinterspersed piezoelectric elements 304 and azimuthal springs 306. Thepiezoelectric elements 304 may each be a stack of ceramic elements usedin the 3-3 mode with alternating polarity, which is preferred and shownin the figure, or a single piece of 3-1 ceramic, or a striped ceramic.The azimuthal springs may have several forms. Shown in FIG. 19 aresprings 306 in the form of deformed tubes which get their compliancefrom the bending of a curved beam. A simple block of elastomer may alsobe used for the azimuthal spring. Because of their arrangement, thesprings may be considered to be in series with the piezoelectricmaterial.

The azimuthal springs 306 allow substantial radial movement of the ring302 as a whole so that it may be used as the acoustic or resonant massof the acoustic resonant system. Flexible skirts 308 and 310 cap thering 302 on both ends of the ring to seal gas in the space inside thering from the gas outside the ring while accommodating the large(perhaps several millimeters) radial motion of the ring. In oneembodiment, each skirt consists of side plates 312, joining regions 314,flexible cones 316, bulges 318, end region 320, hinge regions 322 and anoptional port 324 and attachment ring 326. The skirt 308 is a compliantmechanism. It is formed from a thin fatigue resistant material, such asa tough plastic or spring steel, that should be stiff enough that theskirt holds its shape without buckling under the acoustic pressuredifference that will be applied between inside and outside portions ofthe alternator, yet thin and flexible enough that the skirt 308 is freeto bend at the hinge 322 regions near the perimeter of the side plates312, and that the cone 316 is free to roll or unroll to accommodate thecontractions or expansions of the azimuthal springs 306. The side plates312 are relatively rigid. Their rigidity may be enhanced by making themthicker than the other parts of the skirt, reinforcing them by joiningeach of them to a thicker rigid plate, by deforming the side plate intoa domed shape, or by a combination of these methods. Although in thefigure the hinges 322 are shown to be sharp folds in the skirt forclarity of conveying their function and location, the hinge regionsshould be rounded, making a smooth blend to the rest of the skirt, toavoid stress concentrations.

If the piezoelectric element 304 is a stacked ceramic or a stripedceramic and if the skirts 308 and 310 are made from metal, there shouldbe thin additional electrical insulators, not shown in the figure,inserted between the skirts 308 and 310 and piezoelectric elements atthe joining regions so that the skirt does not electrically short thepiezoelectric elements. If the skirt is fabricated from an insulator, nosuch insulation is needed. If the piezoelectric elements are singleplates used in the 3-1 mode with the side facing the skirt at groundpotential, then again, no insulator is needed at the joining region,although an insulator may then be needed between the piezoelectricelements and the azimuthal springs if the springs are metallic.

The bulges 318 in the cones act to relieve what is mostly axial membranestress in the cone caused by the slight change in angle between thecones and the azimuthal springs near the joining region which occurs asthe resonant ring oscillates radially and the side plates and conesrotate to follow the ring motion.

One of the functions of the end region 320 of the skirt 308 is to allowaxial motion of the hinge region necessary because of the rocking motionof the side plates. To some extent this function is also accomplished bythe hinge region itself if it is curved; the radius of curvature canchange to allow the side plate to follow the radial motion of thepiezoelectric elements while remaining attached to the end region. Theamount of the axial motion to be accommodated depends on the angle thatthe side plates make with the axis of the alternator. The axial motioncan be minimized with side plates that are parallel to the axis, eachside plate in the same plane as its corresponding piezoelectric element(the side plates would then be trapezoidal shaped rather than therectangular shape shown in FIG. 19, with the side facing the end region320 longer than the side facing the joining region 314, to accommodatethe shape of the cone 316), although this will maximize the change inangle that needs to be accommodated by the bulges.

This type of resonant ring alternator 300 without spokes may be usedwith any of the examples of oscillatory motion heat engines describedabove, such as thermoacoustic-Stirling, thermoacoustic standing-wave,thermoacoustic cascade, thermoacoustic no-stack, conventional Stirling,free-piston Stirling, and others. When the alternator is used to convertacoustical power to electrical power, the outside faces of the sidewalls, formed by the piezoelectric elements 304, serve as the movablefaces of the alternator. That is, acoustical power acts on the outsidefaces of the side walls, thereby resonating the ring, thereby stressingthe piezoelectric material. To illustrate the use of the piezoelectricalternator with such engines, the thermoacoustic-Stirling engine will beused by example in the following figures.

FIG. 20 shows in cross section a generator 332 with a resonant ringalternator 330 without spokes being used with a single coaxialthermoacoustic-Stirling engine 352, similar to the engine shownpreviously in FIGS. 4A and 4B. In this embodiment, a port in thealternator 330 is not used. The end regions 334 and 336 of the skirts338 and 340 are held apart by a post 342 and pedestal support 344,defining a trapped portion of gas within the alternator (except for aslow leak, not shown, to allow long term pressure equalization betweenthe inside and outside of the alternator). This trapped “back” volume346 is somewhat disadvantageous because radial motion of the ring 348causes pressure swings within the back volume which cause acousticthermal losses on internal surfaces of the alternator 330. On the otherhand, the generator 332 has a simple construction because it containsonly one engine 352.

There are no net vibrations transmitted to the pressure vessel 350 fromthe radial motion of the resonant ring. However, the vertical gas motionin this embodiment is not balanced—the center of mass of the gas movesaxially as the gas goes through its acoustic cycle, which leads to aaxial vibrations of the pressure vessel.

The generator 332 includes a thermoacoustic engine 352 for convertingheat to sound. The thermoacoustic engine 352 includes a hot heatexchanger 354, a cold or ambient heat exchanger 356 and a regenerator358 sandwiched between the heat exchangers. A nacelle 360 surrounds thethermal core, consisting of the heat exchangers and the regenerator. Thenacelle is generally cylindrical and coaxial with the pressure vessel.The nacelle is spaced from the walls of the pressure vessels so as toprovide an annular space. This embodiment is illustrated with a jet pump362 for counteracting Gedeon streaming. This embodiment is alsoillustrated with a secondary ambient or cold heat exchanger 364 at theopen end of the nacelle 360, such that the thermal buffer tube isdefined between the hot heat exchanger 354 and the secondary exchanger364.

As with the embodiment of FIG. 1, the alternator 330 replaces theacoustic or resonant mass of a traditional thermoacoustic resonator.Also, the annular space between the pressure vessel wall and the nacelle360 acts as a lumped acoustic mass. The space between the cold heatexchanger 356 and the end of the pressure vessel acts as a gas spring.This mass-spring sub-system is preferably tuned to have a resonantfrequency well above the operating frequency of the generator 332, suchthat the pressure oscillations at the upper end of the pressure vessel350 are slightly enhanced. In other words, the pressure peaks areslightly higher on the cold side of the thermal core and the pressureminimums are slightly lower on the cold side of the thermal core.

FIG. 21 shows in cross section a vibration balanced configuration of agenerator 370 for using a resonant ring alternator 372 without spokes.It uses two coaxial engines 374 and 376, separated by a gas tightpartition 378, operating 180 degrees out of phase with each other. Theconfiguration of FIG. 21 is generally radially symmetrical, such thatthe partition 378 forms a cylinder, and the inner and outer engines 374and 376 are each generally annular rings. As gas in the inner engine 374moves downward, gas in the outer engine 376 moves upward, and viceversa. By adjusting the power output and the cross sectional areas ofthe inner and outer engines to be nearly the same, the center of massmotion of the gas in the inner and outer engines can be balanced,leading to a cancellation of vibrations transmitted to the pressurevessel 380, for quieter operation of the generator 370 where warrantedby the extra complexity of having dual engines. The vertical motion ofgas in the engines 374 and 376 drives the radial motion of thealternator which in turn generates electrical power. To minimize thevolume that needs to be swept out by the alternator 372, a filler plug382 is placed in the interior region of the alternator to remove excesscompliance of the interior gas.

The inner and outer engines 374 and 376 are constructed similarly to thethermoacoustic engine of the prior embodiments. The inner engine 374includes a hot heat exchanger 384 and an ambient or cold heat exchanger386 sandwiching a regenerator 388. The nacelle is formed by aninner-nacelle wall 390 in the partition 378. A jet pump 392 is providedon the upper, cold, end of the engine 374, and a secondary exchanger 394is provided on the lower end. Likewise, the outer thermoacoustic engine376 includes a hot heat exchanger 396, and an ambient or cold heatexchanger 398, and regenerator 400. The nacelle is formed is formed byan outer nacelle wall 402 and the partition 378. A jet pump is shown at404 and a secondary heat exchanger is shown at 406.

FIG. 22 shows a similar vibration balanced generator 410 configurationwithout a filler plug. Excess volume in the interior of the alternator412 is taken up by the reentrant shape of the pressure vessel 414. Thecylindrical pocket 416 in the pressure vessel may be put to a beneficialuse, for example, holding electronics associated with the generator.

It should also be mentioned that under some conditions it may beadvantageous to use the generators shown herein in the reverse sense topump heat. What has up to now been described as an alternator may bedriven with high voltage oscillatory electrical power. The alternatorthen becomes functionally a loudspeaker that drives pressureoscillations and acoustic power flow within the pressure vessel. Theengines then become functionally heat pumps or refrigerators driven byacoustic power from the loudspeaker. Also, the transducers may be usedas drivers for other application, such as using them as acousticaldrivers.

As a further alternative, a thermoacoustic device with the generalconfiguration of the device in FIG. 1 may have some of itsthermoacoustic cores used as engines, while other stages are used asheat pumps or refrigerators. Such a configuration is show in FIG. 23,wherein the topmost and second to bottom thermoacoustic cores are usedas engines, while the bottommost and second from the top cores are usedas refrigerators.

Flexible Diaphragm Embodiments

Turning now to FIGS. 24-32, the present invention further includesvarious additional piezoelectric transducers or alternators utilizing aflexible diaphragm for converting between acoustical power and electricpower. These transducers may be used with thermoacoustic devices, orform part of such thermoacoustic devices, as was illustrated withprevious embodiments of transducers here, or may be used in otherapplications. FIGS. 24 and 25 illustrate one embodiment of apiezoelectric transducer/alternator 510 utilizing a diaphragm. FIG. 24is a three dimensional drawing of the flexible diaphragmtransducer/alternator 510, while FIG. 25 shows a cross section of FIG.24 in more detail. A flexible diaphragm 520 is used to couple acousticenergy to a fluid in contact with the diaphragm. A “diaphragm” as usedfor the embodiments of FIGS. 24-32 may have the dynamic properties ofboth a thin flexible plate that has a finite stiffness and a membranethat responds to tension. Alternatively, a “diaphragm” may be a membranewith little or no stiffness, which responds to tension but not tobending stress. The diaphragm 520 may be said to have a central portion522 and a perimeter 524. The perimeter may be the outermost edge of thediaphragm and may be circular, polygonal or have other shapes.Preferably, the perimeter 524 lies in a diaphragm plane. In theillustrated embodiment, the central portion 522 and perimeter 524 of thediaphragm both lie generally in the same plane when the diaphragm is inan unstressed or neutral position. In alternative embodiments, thediaphragm is domed such that it has a concave face and an opposed convexface when in an unstressed or neutral position. The diaphragm 520 may bedisplaced such that the central portion moves axially relative to theneutral position. Typically the central portion will oscillate betweentwo positions displaced in opposite directions from the neutralposition. In the illustrated embodiment, displacement of the diaphragmcauses the central portion 522 to move out of the plane containing theperimeter.

A perimeter member 530 generally surrounds the diaphragm 520 and isinterconnected with the perimeter 524 of the diaphragm, therebysupporting the diaphragm. The perimeter member 530 includes at least onepiezoelectric element 532. In the illustrated embodiment, the perimetermember 530 includes a plurality of piezoelectric elements 532 arrangedgenerally in a ring. The perimeter member is interconnected with thediaphragm 520 such that displacement of the diaphragm stresses thepiezoelectric element or elements. The piezoelectric elements 532 maybe, for example, stacks of piezoelectric ceramic. The cross sectionalshape of the piezoelectric elements, looking into the hoop direction, issomewhat arbitrary. They may be rectangular as shown, round, annular, orof any convenient shape. Some consequences of these choices aredescribed below.

In the illustrated embodiment, the perimeter member 530 further includesa plurality of supports 540 which mechanically interconnect thepiezoelectric elements 532 and diaphragm 520 such that movement of thediaphragm stress the piezoelectric elements. As shown, each support hasa keystone portion 542 positioned between adjacent piezoelectricelements 532 and an attachment portion 544 that is attached to theperimeter 524 of the diaphragm.

The perimeter member 530 may also optionally include azimuthal springs,as was shown previously in the transducer 103 of FIG. 5, and shownpreviously in FIG. 16. The analysis of Example 4 holds as well for thisand other embodiments of the flexible diaphragm transducers disclosedherein.

In the illustrated embodiment, the transducer further includes a strutassembly 550, which includes a plurality of upper struts 552 and lowerstruts 554. Axial motion of the transducer 510 is limited by theplurality of struts 552 and 554. Each strut has one end affixed to thediaphragm 520 near the perimeter and another end attached to a fixedpoint, such as an upper mounting flange 558 or a lower mounting flange560. Together, the struts 552 and 554 and the flanges 558 and 560 definea mounting structure. These struts 552 and 554 are stiff in the axialdirection but are relatively compliant to a slight radial motion of thediaphragm 520, supports 540 and the ring of piezoelectric elements. Thestruts are also relatively compliant to a slight twisting motion of thediaphragm out of its plane. In this embodiment, struts 552 and 554 areplaced on both sides of the diaphragm. This arrangement eliminates thepossibility of buckling of the struts.

A preferred application of the transducer 510 is in a thermoacousticdevice wherein the transducer separates a volume of gas into twoportions. For such an application, as well as some other applications,it is preferred that the transducer provide a substantially fluid tightseal such that the diaphragm separates the volumes. In the illustratedembodiment, the upper mounting flange 558 and lower mounting flange 560are designed to be mounted and sealed to a housing or gas container.This may be accomplished in a variety of ways. In the illustratedembodiment, the flanges 558 and 560 each include a mounting surface andan O-ring groove 562 for mounting and sealing to a housing.

In this embodiment, fluid motion between the struts 552 is blocked byhoop stress relieving members 570, which in this embodiment are thinarcs of metal that are concave in the inward radial direction. Themembers 570 interconnect the struts 552 and may be formed integrallytherewith or formed separately and attached to the struts. Similarmembers may block fluid motion between the lower struts 554.

As will be clear to those of skill in the art, a diaphragm mayexperience hoop stress near its perimeter when the central portion isdisplaced from its neutral position. The amount of hoop stress dependson the characteristics of the diaphragm and the amount of displacement.In some embodiments of the present invention, provision is made toreduce or relieve hoop stress in the diaphragm. In the illustratedembodiment, radial slots 580 in the perimeter of the diaphragm 520relieve hoop stress during the operation of the transducer. As shown,these slots 580 are positioned between the supports 540. Optionally, theslots may be eliminated. As a further alternative, radial corrugationsmay be provided in the diaphragm.

In the embodiment of FIGS. 24 and 25, the perimeter member 530 may besaid to generally define a transducer plane, with the perimeter memberand the diaphragm disposed generally in the plane. The perimeter member530 may also be said to surround a central area, with the diaphragmbeing disposed in this central area.

The diaphragm 520 is able to flex in a direction generally perpendicularto the transducer plane, in the axial direction. When the transducer isused for the generation of electrical power from acoustical power thatmay be derived, for example, from a thermoacoustic-Stirling engine, theacoustic oscillating pressure differential between the two sides of thediaphragm causes the diaphragm 520 to flex and undergo oscillatorymotion in the axial direction. The diaphragm in turn applies anoscillating radial (inward) force on the supports 540 at twice thefrequency of the sound. The oscillating radial force is accompanied by aslight oscillating radial motion of the supports. Thus, acoustic power,in the form of oscillating pressure differential across the diaphragmand oscillatory volume swept out by the diaphragm, is converted intomechanical power in the form of oscillating radial forces on, and motionof, the supports. The oscillating radial force on the supports causesoscillatory compressive stress and strain on the piezoelectric members532 which convert the applied mechanical power into electrical power. Inthis embodiment of the flexible diaphragm transducer when used as analternator, output voltage and current depend quadratically on the axialdisplacement of the diaphragm. For sinusoidal pressures and motion ofthe diaphragm at frequency f, output voltage and current are generatedat 2f.

The transducer may be used in the reverse direction, convertingelectrical power in the form of voltage and current at frequency 2f intoacoustic power at frequency f through the process of parametricexcitation.

As is well known in the art, it may be advantageous to apply a steadybias stress to the transducer. This may be done, for example, by heatingor cooling the diaphragm when assembling the piezoelectric ring to thetransducer, to apply a bias compressive or tensile stress on thepiezoelectric ring, respectively, and a bias tensile or compressivestress on the diaphragm, respectively, after assembly when the diaphragmand piezoelectric ring come to the same temperature, for a diaphragmwith a positive thermal expansion coefficient.

The stresses in the diaphragm used in various embodiments of the presentinvention may limit the power to which the transducer may be operated,even with the use of high strength materials. A high stress regionoccurs near the rim of the diaphragm, such as where the diaphragm 520joins with supports 540 in FIGS. 24 and 25. At full deflection, thediaphragm simultaneously experiences high tensile stress and bendingstress. As is well known (e.g. Roark's Formulas for Stress and Strain,Young and Budynas, Seventh Ed., McGraw-Hill, Table 8.9), the peakbending stress of a bent cantilever beam is increased by tension appliedalong the cantilever. The tension causes the bending of the beam to beconcentrated in a small region near the support. For an imposed slope atthe free end of the cantilever and a given tensile force, thinnercantilevers can have higher peak bending stresses than thicker ones, theopposite of the case of a given bending angle and no tensile force,because as the beam becomes thinner the region of bending becomes moreconcentrated near the support. A similar effect occurs in the diaphragmnear the supports 540. Some embodiments of the present invention includevarious techniques to control this curvature, as will be describedherein.

A first of these curvature control techniques uses to advantage theacoustic pressure on the transducer. For use in a thermoacoustic device,such as shown for example in FIG. 1 or 23, the transducer may be usedadvantageously at a frequency where it presents a mass-like acousticreactance to the thermoacoustic engines and/or chillers. Likewise,thermoacoustic engine and chiller stages present a spring-like acousticreactance to the transducer. Combinations of engines, chillers andtransducers can thus form a compact resonant system, and thethermoacoustic device operates at the frequency of one of theseresonances. Under these conditions, at the phase of the cycle where thediaphragm has deflected maximally downward, say, the main reactivecomponent of the acoustic pressure will be positive below the diaphragmand negative above the diaphragm, so as to accelerate the diaphragmupward. Even though the deflection of the diaphragm will be greatestnear its center, this net pressure difference, positive below thediaphragm, will push upward on the diaphragm fairly uniformly at allpositions because the wavelength of sound in a thermoacoustic generatorwill generally be much larger than the largest dimension of thediaphragm. The net upward pressure presses back on the diaphragm nearthe rim to reduce the increased curvature that is caused by the tensionin the diaphragm. With the proper choice of diaphragm size, maximumtension in the diaphragm, and acoustic properties of the gas springformed by the thermoacoustic engine and chiller spaces, the transducercan be designed to have very low bending moment near the support. Theeffect is similar to that in a kettledrum, and can be studied in Morse'sVibration and Sound (ISBN 0-88318-287-4, reprinted by the AcousticalSociety of America, pp. 193-209). As described in the next fewparagraphs, if the stiffness of the diaphragm is ignored and instead thediaphragm is considered to be a membrane of zero stiffness, with atension that increases as a function of deflection, the membrane may bedesigned to have zero slope at its contact to the rim following thetheory in Morse. Under these conditions, the transducer diaphragm, suchas 520, with its actual plate stiffness, will have minimal bendingstress and the transducer can be used at high power.

The diaphragm may be treated as if it were approximately a circularmembrane with an effective radius a. As will be clear to those of skillin the art, the illustrated membrane is a generally polygonal shapedmember. The effective radius a is the radius of a circular membrane thatperforms equivalently to the polygonal membrane. The radius a will beapproximately equal to the average radius at which the supports attachto the polygonal membrane. The diaphragm has a mass per unit areaσ_(D)=ρ_(D)τ, where ρ_(D) is the mass per unit volume of the materialwith which the diaphragm is constructed and τ is the diaphragmthickness. The diaphragm is also subjected to a time varying membranestress which can be approximated by a tension, the tensile force perlength of edge, given by T=F_(R)/(2πa)+T₀, where, as was shownpreviously in Example 4, is the arithmetic sum of the radial timedependent tensile forces that the piezoelectric ring and supports applyto the diaphragm, and where T₀ is an optional time independent biastension, present when the diaphragm is in equilibrium, which may betaken to be zero if no bias is used. Under these conditions, a membranemay have transverse wave motion with a time dependent speed c given byc²=T/σ_(D). With approximately fixed boundary conditions at the rim ofthe membrane, the displacement of the membrane from equilibrium forradial (azimuthally symmetric) modes of vibration is given byY=A[J ₀(2πυr/c)−J ₀(2πυa/c)] cos(2πυt),  (26)where J₀ is the zero order Bessel function of the first kind, r is theradial coordinate, v is the frequency of motion, t is time, andA[1−J₀(2πυa/c)] is the amplitude of motion at the center of themembrane. Here, an approximation has been made in treating the membraneas if it has a mode shape that varies instantaneously through itsdependence on the time varying speed c. As shown in Morse, the allowedfrequencies of motion of a kettledrum are given by solutions to theequation $\begin{matrix}{{{J_{0}(w)} = {{- \frac{\chi}{w^{2}}}{J_{2}(w)}}},} & (27)\end{matrix}$where w=2πυa/c, χ=πρ₀c_(a) ²a⁴/(V_(o)T), V₀ is the effective volume of agas spring that is compressed by the membrane motion, ρ₀ is the meandensity of the gas that is compressed, c_(a) is the adiabatic soundspeed in the gas, and J₂ is the second order Bessel function of thefirst kind.

In a thermoacoustic device such as that shown in FIG. 1 or 23, thetransducers partition the housing containing the working gas intoseveral effectively closed mean volumes. Volumes may be consideredenclosed by either the walls of the rigid housing containing the workinggas, or by nodal surfaces of the acoustic velocity, which aremathematical surfaces within the gas where the acoustic velocity isnearly zero. In the balanced thermoacoustic devices of FIG. 1 or 23, forexample, there is a nodal surface about midway between the twoalternators. Thus, four effectively closed mean volumes may be defined:the gas filled volume above the upper alternator, the gas filled volumebelow the upper alternator but above the nodal surface at about themid-plane of the housing, the gas filled volume below the nodal surfacebut above the lower alternator, and the gas filled volume below thelower alternator. The effective volume V₀ of the gas spring compressedby the motion of each alternator in such a thermoacoustic device canthen be approximately given by, where and are the two nearesteffectively closed mean volumes of gas on either side of an alternator.Alternatively, a more accurate estimate of the effective volume V₀ ofthe equivalent gas spring that is compressed by the transducer motioncan be made from a thorough acoustic model of the thermoacoustic device,in which case V₀=[γP₀/(2πυ)]Im(U/Δp), where γ=c_(P)/c_(V) is the ratioof specific heats of the gas at constant pressure and constant volume,P₀ is the mean gas pressure, Δp is the complex acoustic pressuredifference across the diaphragm, U is the complex volumetric velocityswing of the diaphragm, and Im denotes the taking of the imaginary partof a complex quantity.

The first diaphragm curvature control technique consists of selecting χsuch that membrane tension and gas pressure forces alone cause thediaphragm to have near zero slope at its rim, imposing near zero bendingmoment on the supports at the rim, and the diaphragm therefore receivingnear zero bending moment reaction back from the supports. The parameterχ=πρp₀c_(a) ²a⁴/(V_(o)T) is a measure of the relative importance of thecompression of the gas to the tension in the membrane as restoringforces on the membrane. At the best value of χ of 14.7, the firstsolution of Eq. (27) for w, when substituted into Eq. (26), gives adeflection Y for the first radial mode of the membrane that has nearzero slope ∂Y/∂r at r=a, even for a diaphragm that is assumed to havezero stiffness. There is therefore no concentration of curvature at therim of a true diaphragm with finite stiffness. For values of χ in therange of 7 and 34, the imposed slope of the membrane near its rim isreduced to 50 percent of the slope imposed without the use of thiscurvature control technique (equivalent to χ=0). It is preferred that adiaphragm be designed to fall In this range. For values of χ in therange of 11 and 19, the imposed slope of the membrane near its rim isreduced to 20 percent, which is more preferred, and for values of χ inthe range of 13 and 17, the imposed slope of the membrane near its rimis reduced to 10 percent, which is even more preferred. Over the range7≦χ≦34, the peak bending stress at the rim of a true finite stiffnessdiaphragm is at least halved where, like the cantilever subjected tosimultaneous bending and tensile forces, the diaphragm must make thesharp transition from the imposed slope of the membrane relatively nearthe rim to the essentially zero slope boundary condition right at therim.

As will be clear to those of skill in the art, a diaphragm with finitestiffness will always have zero slope right at the rim because of theclamped boundary condition. However, the slope transitions rapidly to anonzero slope a short distance from the perimeter. Using the aboveapproach to curvature control, wherein the diaphragm is treated ashaving no stiffness, leads to a finite stiffness diaphragm that has theslope imposed a short distance from the perimeter being reduced oreliminated, thereby reducing a concentration of bending stress. Thiscurvature control technique puts constraints on the size of thediaphragm, which may be inconvenient, but it has the advantage ofmanufacturing simplicity because it does not require additional partsfor the control of diaphragm curvature.

A second diaphragm curvature control technique for use with the presentinvention is to vary the thickness of the diaphragm. The diaphragm maybe thickened in regions where the curvature would otherwise be high,such as near the rim, to locally reduce the bending stress. However, thediaphragm also becomes stiffer where it becomes thicker. In order tocouple a certain amount of acoustic power to or from the fluid, it isnecessary for the diaphragm to sweep out a certain volume during itsflexure. Stiffening the diaphragm in one region will make it moredifficult sweep out that volume without exceeding the maximum allowablestress in another region of the diaphragm.

Another embodiment of a flexible diaphragm transducer is shown at 600 inFIG. 26 to illustrate some alternative design choices that may be madeto fit a given application. In this embodiment, the perimeter member 608includes piezoelectric elements 610 and 612 that are grouped into twonominally circumferential rings on either side of the diaphragm 620.Also, the rings of piezoelectric elements 610 and 612 are placed insidethe outer edge of the diaphragm, which is an advantage when making atransducer with the lowest possible maximum radial dimension that has agiven size diaphragm. The shape of the supports 640 is changed relativeto that in FIGS. 24 and 25 to accommodate this alternate placement ofthe piezoelectric elements 610 and 612. Struts 650 are placed on onlyone side of the diaphragm, which is allowable as long as they can bedesigned to not buckle during the part of the acoustic cycle when theyexperience compressive stress in the axial direction. The struts 650 inthis embodiment have also been connected to the supports 640 rather thanto the diaphragm. The ends of the struts opposite the diaphragm andsupports provide the mounting surface for the transducer, and may bejoined, for example, to a mounting flange 660, or some other structure,as determined by the application. For example, the ends of the strutsopposite the diaphragm and supports may be joined to a component of athermoacoustic engine, for the generation of electrical power, withoutthe flange 660. Alternatively, one flexible diaphragm transducer may beattached to another flexible diaphragm transducer at the ends of theiraxial struts opposite their respective diaphragms, to couple with eitherdipole or monopole acoustic fields depending on whether the twotransducers are operated in phase or in opposite phase, respectively.

Since the desired compressive stress in the piezoelectric elements isessentially in parallel mechanically with the hoop stress in thediaphragm, hoop stress relieving radial slots 680 in the diaphragm mayextend over much of the diaphragm so as to deliver additionalcompressive stress to the piezoelectric elements for a given amount ofdiaphragm motion. Such long radial slots reduce the amount of mechanicalenergy stored in the diaphragm during its deflection, and improve theratio of energy coupled between acoustic and electrical forms to thatstored in the transducer. This can be a major advantage in some sonarapplications if the piezoelectric elements are made from high couplingfactor piezoelectric materials, such as the single crystal piezoelectricmaterials currently under development in many laboratories. Fluidpassage through the radial slots 680, the spaces between the supports640, and the spaces between the struts 650 is blocked in this embodimentby a relatively compliant elastomer seal 670, such as a rubber,silicone, or polyurethane adhesive, forming an acoustic seal. A slightdisadvantage of long radial slots 680 in the diaphragm 620 is thecreation of a stress concentration at the end of the slot 680. This canbe mitigated with a teardrop or circular cutout at the end 685 of theslot 680 as shown in FIG. 27A. The cutout can also be filled withelastomer seal 670 as shown in FIG. 27B. Another end condition for aradial slot 681, shown in the detail FIG. 27C, is a “J” end condition686, which has the advantage of more fully supporting the slot fillingelastomer 671, shown in detail FIG. 27D, against stresses caused by theacoustic pressure.

A third diaphragm curvature control technique is illustrated in theembodiment of FIG. 26. Two curved guiding surfaces 690 are placed oneither side of the diaphragm 620, approximately tangent to the diaphragmsurface. These surfaces may be an integral part of the supports 640, asin FIG. 26, or they may be the surface of separate parts. Theinteraction of curved guiding surface 690 and the diaphragm 620 has somesimilarities to that of a rolling contact bearing. It would thereforeusually be advantageous to pick combinations of materials for theseparts that minimize the effects of surface fatigue, wear, or frettingfatigue. For example, the diaphragm 620 could be made from a fatigueresistant high strength steel or titanium, while the curved guidingsurface 690 could be made from high strength steel, brass, bronze,Babbitt metal, or a tough, low friction, wear resistant plastic such asa member of the Rulon® family.

Both linear and quadratic (or higher order nonlinear) response may beobtained from a flexible diaphragm transducer that has separateelectrical connections for the piezoelectric material on the two sidesof the plane of the diaphragm. For the transducer of FIG. 26, thepiezoelectric elements 610 on one side of the diaphragm may be wiredtogether into a first bank of elements, while the elements 612 on theother side of the diaphragm may be wired together into a second bank ofelements. For the embodiment of FIGS. 24 and 25 each of thepiezoelectric elements 532 may be modified into either two separateelements on the top side and bottom side of the diaphragm, or eachelement may have split electrodes and electrical connections for the tophalf and the bottom half. In either case, portions of piezoelectricelements 532 on the top half and bottom half of the plane of thediaphragm may be wired together into first and second banks ofpiezoelectric elements.

The operation of a flexible diaphragm transducer to obtain both linearand quadratic response is shown in FIGS. 28A-28E. The five frames show asimple embodiment of a flexible diaphragm transducer in cross section atfive phases of an oscillation, that may serve to illustrate theoperation of several of the embodiments. The motion of the transducer isgreatly exaggerated for clarity. The simple transducer consists of aflexible diaphragm 720 joined to an upper ring of piezoelectric material730 and joined to a lower ring of piezoelectric material 735. A simplemeans for mounting a flexible diaphragm transducer to a fixed structure760 is a compliant elastomer ring 750. FIG. 28C shows the transducer atits equilibrium or neutral position, wherein the central portion andperimeter of the diaphragm 720 are disposed in a common plane. FIG. 28Bshows the diaphragm 720 flexed upward moderately far. In response to thefinite stiffness of the diaphragm (and optionally in response toreaction forces on curved guiding surfaces 690 of FIG. 26 if they arepresent), a twisting moment is applied to the rings 730 and 735. Thetwisting moment results in the circumference of the upper ring beingincreased and the circumference of the lower ring being decreased,indicated by the arrows. Thus the tensile stress of the upper ring isincreased, toward tension, and the tensile stress of the lower ring isdecreased, toward compression. For these relatively small deflections,the flexible diaphragm transducer operates similarly to a bimorph. FIG.28A shows the situation at a larger deflection than in FIG. 28B. Thedeflection in this case is taken to be high enough that the quadraticcompression of both rings overwhelms the linear twisting effect, andboth rings are compressed relative to their equilibrium positions. Thishigher deflection operation of the flexible diaphragm transducer isdissimilar to the operation of a bimorph, and can be an advantage athigher powers. It places both rings of piezoelectric material undercompression, where they are strong, rather than tension, where they areweak. FIGS. 28D and 28E show the result of diaphragm deflections in theopposite direction.

FIG. 29 shows the open circuit output voltage V from the upper bank ofpiezoelectric elements and for the lower bank of piezoelectric elementsas a function of the membrane deflection x at its center, when thetransducer is used as an alternator to generate electrical power. Here Vis taken to be positive for stress that is compressive relative to theequilibrium stress, and x taken to be positive upward. The x axis islabeled with the deflections of the corresponding five frames of FIGS.28A-28E. Small amplitude oscillations can be represented by the sequencec-b-c-d-c in FIGS. 28 and 29. Large amplitude oscillations can berepresented by the sequence c-b-a-b-c-d-e-d-c. It can be seen that eachbank of piezoelectric elements 730 and 735 present a linear responsenear the equilibrium position c. For small sinusoidal displacementamplitude oscillations at frequency f, the dominant frequency componentof the voltage will also be f. For larger sinusoidal displacementamplitude oscillations at f, the voltage will have f and 2f components.The linear component can be emphasized by designing the transducer witha thicker, and therefore stiffer, diaphragm and piezoelectric elementsthat are placed relatively close to the plane of the diaphragm. Thequadratic component can be emphasized with a thinner, and therefore lessstiff, diaphragm and piezoelectric elements that are placed further fromthe plane of the diaphragm.

When the transducer is used in the opposite direction, as an acousticsource, the voltage V of FIG. 29 may be interpreted as the drivingvoltage needed to be supplied to each piezoelectric element bank toachieve a deflection x. The unloaded diaphragm deflection can be madelinearly proportional to an input voltage proportional to x, even withthe transducer driven into its nonlinear regime, by including a circuitthat converts the input voltage proportional to x into the upper andlower bank driving voltages using the functions of FIG. 29.

A fourth diaphragm curvature control technique is illustrated by thecutaway drawing of an embodiment of a transducer 810 of FIG. 30. Thistechnique uses a plurality of springs 890 and 895 that are attached tothe diaphragm 820 at one end and to a relatively fixed position, such asa rigid support structure, or as shown, to supports 840. The springs mayhave any convenient form. In the illustrated embodiment, one set of leafsprings 890 is provided above the diaphragm and another set 895 isprovided below the diaphragm. More than one layer of leaf springs may beplaced on either side of the diaphragm. Other spring forms, such as coilsprings, or a layer of compliant low-loss elastomeric material placedbetween the diaphragm and a relatively fixed position, are possible aswell. The diaphragm curvature is controlled by the distribution of thesprings on the diaphragm and their spring constants. The bending of thediaphragm is similar to the bending of a beam on an elastic foundation.Like the technique of FIG. 26, this technique allows the diaphragm to bemade thin, to reduce the peak bending stress for a given amount ofdisplacement of the diaphragm, without suffering the increase in bendingstress that would occur without the control of the curvature in thepresence of high tensile forces.

The transducer 810 is held in place axially by a plurality of struts 850attached to the supports 840 and/or the diaphragm 820. A plurality ofpiezoelectric elements 830, which for example may be in the form of astack of piezoelectric ceramic used in its 33 or its 31 mode, isattached to the supports 840. The plurality of piezoelectric elementsmay be electrically connected into a single bank, or as two separatebanks of elements, one above the plane of the diaphragm 820 and onebelow, as described earlier. The struts 850 are stiff axially but arerelatively compliant to radial motion of the supports, diaphragm, andelectroactive material that occurs in response to the axial flexing ofthe diaphragm. The struts in this embodiment are shown as portions of agenerally tubular piece 855. Together, the struts 850 and the piece 855serve as a mounting structure. The generally tubular piece 855 includesa generally circular cylindrical portion 860 which nominally has nomotion and can be used as the mounting surface of the transducer. Smoothtransitions 865 are made to blend the generally polygonal cylinderformed by the ring of struts 850 to the circular cylindrical portion860. Gaps between the struts relieve hoop stress during radial motion ofthe diaphragm rim, supports, and piezoelectric elements. These gaps arefilled with a filler 870 to seal the gaps against acoustically drivenfluid flow. The filler 870 is compliant to compression in the hoopdirection, such as an elestomeric material (as drawn) or a thin arch ofmetal, as in previously described embodiments. Similarly, radial slotsmay be placed in the diaphragm to reduce hoop stress in the diaphragm,and these slots may be filled with a filler for sealing againstacoustically driven fluid flow. These hoop stress reduction techniquesin the supports and diaphragm allow for more of the developed hoopstress being concentrated on the piezoelectric elements and thus improvethe energy coupling factor between the piezoelectric elements and theacoustical field. Alternatively, a transducer may be assembled withouthoop stress reduction techniques.

Another method of linearizing the response of a flexible diaphragmtransducer is shown in the simplified embodiment of FIGS. 31A-31C. FIG.31B shows this transducer 910 in its equilibrium configuration. Insteadof the flat diaphragms of the previous embodiments, the diaphragm 920 isgenerally dome shaped at equilibrium, in the absence of sound. Putanother way, the diaphragm 920 has a convex face and opposed concaveface, with these faces also defining the faces of the transducer.Surrounding diaphragm 920 is a piezoelectric ring 930. It may consist ofa simple single ring of piezoelectric material, such as a piezoelectricceramic, or the piezoelectric ring 930 may be taken to represent aplurality of supports and piezoelectric elements, as in previouslydescribed embodiments. The transducer 910 may be supported axially inany number of ways, including the other approaches described herein. Inthe illustrated embodiment, a compliant support 950, which may be anelastomeric material, is illustrated between the piezoelectric ring 930and a fixed structure 980. The compliant support 950 is relatively stiffto axial motion, but allows for a slight radial motion of thepiezoelectric ring 930. The radial motion of the piezoelectric ring,shown in FIGS. 31A and 31C, has been exaggerated for clarity. The domeddiaphragm may be used in place of the flat diaphragm in any of the priorflexible diaphragm transducer embodiments, and may have slots, strutsand/or supports as discussed previously.

The three frames of FIG. 31 show the operation of the domed diaphragmtransducer over three phases of an acoustic cycle. The cycle may bevisualized by the sequence of frames b-a-b-c-b, and so on. Frame a,shown in FIG. 31A, shows the phase when the diaphragm is deflectedupward. If the transducer is operated as an alternator, convertingacoustical power into electrical power, the membrane stress in thediaphragm in frame a becomes more tensile. The diaphragm pulls inward onthe piezoelectric ring, causing the stress in the piezoelectric ring tobecome more compressive, and the piezoelectric ring generates a voltagewith a particular sign. As the oscillation proceeds through frame b,shown in FIG. 31B, to frame c, shown in FIG. 31C, the stresses reverse.The membrane stress in the diaphragm becomes more compressive, thediaphragm pushes outward on the piezoelectric ring, the stress in thepiezoelectric ring becomes more tensile, and the piezoelectric ringgenerates a voltage with the opposite sign. Thus the transducer has alinear response component. Acoustic oscillations at frequency f generatea voltage that is also at frequency f. At high acoustic drive levels,the response may switch over to a nonlinear quadratic response,generating a voltage response component at frequency 2f, which may beadvantageous in power generation applications. For operation of thetransducer as a projector of sound, converting electrical power intoacoustical power, the relative phase of the stresses with respect to thediaphragm displacement are reversed.

Any generally dome shaped profile for the diaphragm, such as parabolic,spherical or elliptical for example, will provide a linear responsecomponent to the transducer. But, the profile in FIG. 31B illustrates adiaphragm profile that may be designed to provide another curvaturecontrol technique at the rim or perimeter of the diaphragm. For purposesof description, the equilibrium position of the diaphragm in frame b ofFIG. 31B is shown displaced generally upward from the plane defined bythe rim of the diaphragm. In other words, the central portion of thediaphragm is displaced above a plane defined by the perimeter, which mayalso be the transducer plane. Operation of the transducer as analternator, to convert acoustic power into electrical power, will bedescribed. When, during the oscillation, the acoustic pressure causesthe diaphragm to be displaced further upward as in frame a, the bendingstresses near the rim are compressive on the top of the diaphragm andtensile on the bottom. These bending stresses are made more severe bythe desirable membrane tensile stresses that are generated by thediaphragm deformation to pull inward on the piezoelectric ring 930.These bending stresses can be mitigated by the concave upward region922. Tension causes concave region 922 to straighten out in frame a,creating compensating bending stresses that are tensile on the top ofthe diaphragm and compressive on the bottom. This allows the alternatorto be used to higher amplitude, even into the nonlinear, quadraticregime, for the generation of higher electrical power. On the other halfof the cycle, illustrated in frame c, the acoustic pressure causes thediaphragm to be displaced downward, the desirable membrane stresses inthe diaphragm become more compressive, and the compressive stress pushesout on the piezoelectric ring 930. During this phase of the cycle, theconcave upward region 922 “unfolds,” becoming more straight, whichdecreases the tendency of the diaphragm to buckle under the compressiveloading, which is also desirable.

Referring now to FIG. 32, another version of a flexible diaphragmtransducer is shown at 1010. This design is similar to earlierembodiments, but includes radial slots 1022 in the diaphragm 1020 andaxial slots 1024 in the struts 1026, with the slots all being sealed byarches 1028. The version illustrated in FIG. 32 may be machined out of asingle or multiple pieces of material, with the arches integrallyformed, or the arches may be separately formed and then attached such asby brazing. In any of the embodiments, the hoop stress relieving radialslots may be eliminated, depending on the application and design.

FIGS. 33A-33C provide detailed views of how an arch may be formed toseal the radial and/or axial slots in a transducer according to thepresent invention, such as the transducers of FIG. 26 or 32. For thetransducer of FIG. 26, the arch would replace the filler material. FIGS.33A and 33B provide prospective views of a radial arch where it joins anaxial arch. As shown, the arch has a concave and a convex side. FIG. 33Cillustrates the transition area between the two arches.

The various embodiments of transducers discussed herein may be used inthermoacoustic devices, such as those shown in FIGS. 1 and 23, and mayalso be used in other applications requiring a transducer to convertbetween acoustical power and electrical power. The flexible diaphragmtransducers may provide a moving mass that serves as a substantialportion of the resonating mass in a thermoacoustic device, as wasdiscussed with earlier transducer embodiments.

The various embodiments of transducers discussed herein have beendescribed as utilizing piezoelectric materials or elements. Examples ofsuitable piezoelectric materials are piezoelectric ceramics, such asmembers of the lead zirconate titanate (PZT) family, including PLZT((Pb, La)(Zr,Ti)O₃), single crystal materials such as PMN-PT(Pb(Mg_(1/3)Nb_(2/3))_(x-1)TiO₃), piezoelectric polymers such as PVDF(polyvinylidene fluoride) and piezoelectric copolymers. As will be clearto those of skill in the art, these piezoelectric materials may bereplaced with other transduction materials, members of the general classof “electroactive” materials (which include piezoelectric materials),for example electrostrictive materials, such as PMN-PT ceramic, ormagnetostrictive materials, such as Terfenol-D or Galfenol.Electrostrictive and magnetostrictive materials may be used to advantagewith optional bias electric or magnetic fields, respectively. Theremainder of the transducer may be constructed of various materials, aswill be clear to those of skill in the art. The diaphragm, struts,arches, and supports, may, for example, be formed of materials thatexhibit a sufficiently high fatigue endurance limit, such as steel,particularly carbon steel or stainless steel, titanium and its alloys,nickel and its alloys, beryllium copper, phosphor bronze, brass,polymers, or composite materials such as carbon fiber composites orfiber reinforced rubber. Important exemplary dimensions for thediaphragm, struts, arches, and supports of a 4 kW transducer forthermoacoustic application, similar to that shown in FIGS. 24 and 25,and made from PH 15-7 Mo precipitation hardening stainless steel(supports made from 13-8 Mo stainless steel), are diaphragm radius of7.1″, diaphragm thickness of 0.040″, strut thickness of 0.025″, strutheight of 1″, arch thickness of 0.002″, arch width of 0.20″, and supportheight of 1″. The metal parts of this exemplary transducer are joinedtogether with vacuum nickel braze. The eighteen piezoelectric stacks ofthis exemplary transducer are each made from 54 Navy Type III PZTceramic elements, 1″ high, 0.30″ wide, and 0.039″ thick, poled acrossthe thickness. As will be clear to those of skill in the art, theelectroactive materials used in some embodiments of the presentinvention will be subjected to high temperatures. As such, electroactivematerials and assembly methods suited for high temperature applicationsare preferred for these embodiments.

Applications of Thermoacoustic-Piezoelectric Generators

As will be appreciated by those of skill in the art, the generatorsillustrated herein may be used in a wide variety of applications. Thefollowing list provides some exemplary applications.

-   -   Modular, distributed thermoacoustic piezoelectric power plants,        some with integral chillers, for powering ships or aircraft.    -   A chiller for a chemical protection suit that also generates        electricity and is therefore able to reduce the overall weight        carried by the user by reducing the number of batteries to be        carried in other electronic systems.    -   A thermoacoustic piezoelectric generator/chiller can potentially        be made substantially lighter than an equivalent internal        combustion engine, conventional generator, and vapor compression        refrigerator combination. Airlifted thermoacoustic auxiliary        power units for heating, cooling and electricity generation may        therefore be attractive because of weight savings.    -   Because a thermoacoustic piezoelectric generator does not need        air (as required by gas turbine, diesel, or gasoline engines),        thermoacoustic generators may be considered for space        applications.    -   A thermoacoustic piezoelectric generator and chiller/heat pump        unit can be used to generate electricity, winter heating and        summer cooling for use in the cab of a truck for the comfort of        the driver during mandatory rest stops, and for preheating of        the main diesel engine and fuel system for easier start up. The        unit may be powered by combusting diesel fuel. This allows the        main diesel engine of the truck to be turned off rather than        left at idle as is commonly practiced at rest stops. This would        have beneficial impacts on fuel usage, wear on the main engine,        exhaust emissions and noise pollution.    -   Waste heat from the exhaust of trucks or cars may be used to        thermoacoustically generate air conditioning and electricity        that may be used to provide for additional motive power in a        hybrid vehicle.    -   A thermoacoustic piezoelectric generator may be used to generate        additional electricity as a bottoming cycle from the waste heat        of fuel cells, particularly solid oxide fuel cells which run        with a high (850° C.) exhaust temperature.    -   A thermoacoustic piezoelectric generator/chiller may be used to        refrigerate the cargo of a truck while generating the        electricity needed to run fans to circulate air in the        refrigerated compartment. The unit may be powered by combusting        diesel fuel.    -   The projected compactness of a thermoacoustic piezoelectric        generator suggests that it may be useful as an automobile power        plant, saving weight over a conventional gasoline engine.    -   Thermoacoustic piezoelectric power generation may be used as an        environmentally benign source of hydrogen. Solar power can be        used to power a thermoacoustic piezoelectric generator, perform        hydrolysis on water, and produce hydrogen gas. The generator may        also contain a thermoacoustic cryogenic liquefier for the        hydrogen gas. It is therefore possible to create liquid hydrogen        out of nothing more than sunlight and water.

While the invention has been particularly shown and described withreference to preferred embodiments thereof, it will be understood bythose skilled in the art that various alterations in form and detail maybe made therein without departing from the spirit and scope of theinvention. In particular, the specific design of the power generatorincluding heat and coolant sources, shape and design of the resonators,nacelles, heat exchangers, and regenerators can vary widely within theguidelines disclosed herein. This also applies to the piezoelectricalternator design, which can vary so long as it is capable oftranslating pressure oscillations into piezoelectric stress to generatepower. Other variations will be clear to those of skill in the art. Itis the following claims, including all equivalents which define thescope of the present invention.

1. An electroactive transducer for converting between acoustical power,consisting of pressure and velocity, and electrical power, consisting ofpotential and current, the transducer comprising: a diaphragm having aperimeter and a central portion, the diaphragm having a neutral positionand at least one flexed position wherein the central portion isdisplaced axially relative to the neutral position, the perimeter andcentral portion disposed generally in a common diaphragm plane when thediaphragm is in the neutral position; and a perimeter member includingat least one electroactive element, the perimeter member generallydefining a transducer plane and surrounding a central area, theperimeter member being mechanically coupled to the perimeter of thediaphragm such that displacement of the central portion from the neutralposition to the flexed position stresses the electroactive element. 2.The electroactive transducer according to claim 1, wherein: thediaphragm plane is generally coextensive with the transducer plane andthe diaphragm is disposed in the central area defined by the perimetermember such that the perimeter member generally surrounds the diaphragm.3. The electroactive transducer according to claim 1, wherein: the atleast one electroactive element comprises a plurality of electroactiveelements disposed generally in a ring.
 4. (canceled)
 5. Theelectroactive transducer according to claim 3, wherein: the perimetermember further comprises a plurality of supports, each support having aportion disposed between two adjacent electroactive elements and anotherportion interconnected with the perimeter of the diaphragm. 6.(canceled)
 7. The electroactive transducer according to claim 3,wherein: the plurality of electroactive elements is disposed such that aportion of each element is above the diaphragm plane and another portionof each element is below the diaphragm plane.
 8. The electroactivetransducer according to claim 1, wherein: the diaphragm has a pluralityof radial slots defined from the perimeter and extending partially intothe central portion. 9-10. (canceled)
 11. The electroactive transduceraccording to claim 1, further comprising: a mounting structure formounting the transducer, the mounting structure including a plurality ofstruts each having a first end interconnected with the diaphragm. 12-13.(canceled)
 14. The electroactive transducer according to claim 1,wherein: the at least one electroactive element comprises a firstplurality of electroactive elements disposed generally in a first ringand a second plurality of electroactive elements disposed generally in asecond ring, the first ring being disposed above the diaphragm plane andthe second ring being disposed below the diaphragm plane.
 15. (canceled)16. An electroactive transducer for converting between acoustical power,consisting of pressure and velocity, and electrical power, consisting ofpotential and current, the transducer having a first face and an opposedsecond face, the transducer comprising: a diaphragm having a firstconvex face defining at least a portion of the first face of thetransducer and an opposed concave face defining at least a portion ofthe second face of the transducer, the diaphragm having a perimeter anda central portion, the diaphragm having a neutral position and at leastone flexed position wherein the central portion is displaced axiallyrelative to the neutral position; and a perimeter member including atleast one electroactive element, the perimeter member generally defininga transducer plane and surrounding a central area, the perimeter memberbeing mechanically coupled to the perimeter of the diaphragm such thatdisplacement of the central portion from the neutral position to theflexed position stresses the electroactive element.
 17. Theelectroactive transducer according to claim 16, wherein: the perimeterof the diaphragm is disposed in the transducer plane and the diaphragmis disposed in the central area defined by the perimeter member suchthat the perimeter member generally surrounds the diaphragm.
 18. Theelectroactive transducer according to claim 17, wherein: the at leastone electroactive element comprises a plurality of electroactiveelements disposed generally in a ring.
 19. (canceled)
 20. Theelectroactive transducer according to claim 18, wherein: the perimetermember further comprises a plurality of supports, each support having aportion disposed between two adjacent electroactive elements and anotherportion interconnected with the perimeter of the diaphragm.
 21. Theelectroactive transducer according to claim 20, wherein: the diaphragmhas a plurality of radial slots defined from the perimeter and extendingpartially into the central portion, each support being interconnectedwith the perimeter of the diaphragm between the slots.
 22. Theelectroactive transducer according to claim 18, wherein: the pluralityof electroactive elements is disposed such that a portion of eachelement is above the transducer plane and another portion of eachelement is below the transducer plane. 23-29. (canceled)
 30. Athermoacoustic generator, refrigerator or heat pump comprising: ahousing containing a working volume of gas with a pressure; athermoacoustic engine supported in the housing and having a first heatexchanger and a second heat exchanger, the thermoacoustic engine beingoperable to introduce acoustical power into the housing or to removeacoustical power from the housing; and an electroactive transducersupported in the housing, the transducer comprising: a diaphragm havinga perimeter and a central portion that is movable when acted on byacoustical power, the diaphragm having a neutral position and at leastone flexed position wherein the central portion is displaced axiallyrelative to the neutral position; and a perimeter member including atleast one electroactive element, the perimeter member generally defininga transducer plane and surrounding a central area, the perimeter memberbeing mechanically coupled to the perimeter of the diaphragm such thatdisplacement of the central portion from the neutral position to theflexed position stresses the electroactive element.
 31. Thethermoacoustic generator, refrigerator or heat pump according to claim30, wherein the diaphragm divides the housing into a first portion and asecond portion that are generally sealed from each other.
 32. Anacoustic device comprising: a housing having a volume defined therein,the housing being filled with a gas having a mean density ρ₀, and anadiabatic sound speed C_(a); an electroactive transducer at leastpartially disposed in the housing, the transducer comprising: adiaphragm having a perimeter and a central portion, the diaphragm havinga neutral position and at least one flexed position wherein the centralportion is displaced axially relative to the neutral position; and aperimeter member including at least one electroactive element, theperimeter member generally defining a transducer plane and surrounding acentral area, the perimeter member being mechanically coupled to theperimeter of the diaphragm such that displacement of the central portionfrom the neutral position to the flexed position stresses theelectroactive element; the diaphragm dividing the housing such that afirst volume is defined on one side of the diaphragm and a second volumeis defined on the other side of the diaphragm, the diaphragm having aneffective radius a; the diaphragm having a tensile force per length ofedge T, given by T=F_(R)/(2πa)+T₀, wherein F_(R) is the arithmetic sumof all radial time dependent tensile forces that the perimeter memberapplies to the diaphragm and T₀ is a bias tension, present when thediaphragm is in the neutral position; the diaphragm being configuredsuch that χ is in the range of 7 to 34, inclusive, wherein χ is definedasχ=πρ₀ c _(a) ² a ⁴/(V _(o) T) Z wherein V₀ is the effective volume ofthe gas spring compressed by the diaphragm.
 33. The acoustic device ofclaim 32, wherein V₀=1/(V₁ ⁻¹ +V ₂ ⁻¹), wherein V₁ and V₂ are the twonearest effectively closed mean volumes of gas on either side of thediaphragm.
 34. The acoustic device of claim 32, further comprising athermoacoustic engine supported in the housing and having a first heatexchanger and a second heat exchanger, the thermoacoustic engine beingoperable to introduce acoustical power into the housing or to removeacoustical power from the housing. 35-36. (canceled)
 37. A piezoelectrictransducer for converting between acoustical power, consisting ofpressure and velocity, and electrical power, consisting of potential andcurrent, the transducer comprising: a perimeter member including atleast one portion of piezoelectric material, the perimeter memberconfigured such that compression of the perimeter member causescompression of the portion of piezoelectric material, the perimetermember surrounding a central area; a hub disposed in the central area,the hub being movable relative to the perimeter member along an axis;and a plurality of spokes interconnecting the hub and the perimetermember such that relative movement of the hub along the axis compressesthe perimeter member and thereby compresses the piezoelectric material.38-46. (canceled)
 47. A piezoelectric transducer for converting betweenacoustical power, consisting of pressure and velocity, and electricalpower, consisting of potential and current, the transducer comprising: aperimeter support member generally defining a transducer plane, themember surrounding a central area; a hub disposed in the central area,the hub being movable relative to the perimeter support member along anaxis generally perpendicular to the plane; and a plurality ofpiezoelectric bimorph members each having an inner end in mechanicalcommunication with the hub and an outer end supported by the perimetersupport member such that relative movement of the hub along the axisflexes the bimorph members. 48-50. (canceled)
 51. A thermoacousticdevice comprising: a housing containing a working volume of gas with apressure; a piezoelectric transducer separating the housing into a firstarea containing a first volume of gas and a second area containing asecond volume of gas, the transducer comprising a perimeter wall havingat least one portion of piezoelectric material and at least one springin series; a first thermoacoustic core supported in the first area ofthe housing and including a pair of heat exchangers; and a secondthermoacoustic core supported in the second area of the housing andincluding a pair of heat exchangers.